Exponential and Logarithmic Functions and Relations
Then
Now
Why?
In Chapter 2, you graphed functions and transformations of functions.
In Chapter 8, you will:
SCIENCE Mathematics and science go hand in hand. Whether it is chemistry, biology, paleontology, zoology, or anthropology, you will need strong math skills. In this chapter, you will learn mathematical aspects of science such as computer viruses, populations of insects, bacteria growth, cell division, astronomy, tornados, and earthquakes.
Graph exponential and logarithmic functions. Solve exponential and logarithmic equations and inequalities. Solve problems involving exponential growth and decay.
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Tennessee Curriculum Standards ✔ 3103.3.13
Get Ready for the Chapter Diagnose Readiness
1

You have two options for checking prerequisite skills.
Textbook Option Take the Quick Check below. Refer to the Quick Review for help.
QuickCheck
QuickReview
Simplify. Assume that no variable equals zero. (Lesson 61)
Example 1
4 3 5
(a bc ) _ . Assume that no variable equals zero. 2 2
3
1. a a a
Simplify
2. (2xy 3z 2)
3
a 4a 2b 2bc 5c 3
2
(a 3bc 2) _
8 5
24x y z 3. _ 2 8 6
4 2 2
a a b bc c
16x y z
(
2
8r n 4. _ 3 36n t
)
5 3
a b c =_ 6 3 8
Simplify the numerator by using the Power of a Power Rule and the denominator by using the Product of Powers Rule.
1 =_ or b 1c 4 4
Simplify by using the Quotient of Powers Rule.
6 2 4
2
a b c
5. DENSITY The density of an object is equal to the mass divided by the volume. An object has a mass of 7.5 × 10 3 grams and a volume of 1.5 × 10 3 cubic centimeters. What is the density of the object?
Find the inverse of each function. Then graph the function and its inverse. (Lesson 72) 6. f (x ) = 2x + 5
7. f (x ) = x  3
8. f (x ) = 4x
1 9. f (x ) = _ x3
x1 10. f (x ) = _ 2
4
1 11. y = _ x+4 3
Determine whether each pair of functions are inverse functions. 12. f (x ) = x  6 g(x ) = x + 6
bc
Example 2 Find the inverse of f (x ) = 3x  1.
Step 1 Replace f (x ) with y in the original equation: f (x ) = 3x  1 → y = 3x  1. Step 2 Interchange x and y : x = 3y  1. Step 3 Solve for y. x = 3y  1
13. f (x ) = 2x + 5 g(x ) = 2x  5
14. FOOD A pizzeria charges $12 for a medium cheese pizza and $2 for each additional topping. If f (x ) = 2x + 12 represents the cost of a medium pizza with x toppings, find f 1(x ) and explain its meaning.
Inverse
x + 1 = 3y
Add 1 to each side.
x+1 _ =y
Divide each side by 3.
3
_1 x + _1 = y 3
3
Simplify.
Step 4 Replace y with f 1(x ). 1 1 1 1 y=_ x+_ → f 1(x ) = _ x+_ 3
2
3
3
3
Online Option Take an online selfcheck Chapter Readiness Quiz at connectED.mcgrawhill.com. 473
Get Started on the Chapter You will learn several new concepts, skills, and vocabulary terms as you study Chapter 8. To get ready, identify important terms and organize your resources. You may wish to refer to Chapter 0 to review prerequisite skills.
StudyOrganizer
NewVocabulary
Exponential and Logarithmic Functions and Relations Make this Foldable to help you organize your Chapter 8 notes about exponential and logarithmic functions. Begin with two sheets of grid paper.
1
Fold in half along the width.
First Sheet
2
On the first sheet, cut 5 cm along the fold at the ends.
Second Sheet
3
On the second sheet, cut in the center, stopping 5 cm from the ends.
English
Español
exponential function
p. 475
función exponencial
exponential growth
p. 475
crecimiento exponencial
asymptote
p. 475
asíntota
growth factor
p. 477
factor de crecimiento
exponential decay
p. 477
desintegración exponencial
decay factor
p. 478
factor de desintegración
exponential equation
p. 485
ecuación exponencial
compound interest
p. 486
interés compuesto
exponential inequality
p. 487
desigualdad exponencial
logarithm
p. 492
logaritmo
logarithmic function
p. 493
función logarítmica
logarithmic equation
p. 502
ecuación logarítmica
logarithmic inequality
p. 503
desigualdad logarítmica
common logarithm
p. 516
logaritmos communes
Change of Base Formula
p. 518
fórmula del cambio de base
natural base, e
p. 525
e base natural
natural base exponential function
p. 525
base natural función exponencial
natural logarithm
p. 525
logaritmo natural
ReviewVocabulary
4
Insert the first sheet through the second sheet and align the folds. Label the pages with lesson numbers.
domain p. P4 dominio the set of all xcoordinates of the ordered pairs of a relation function p. P4 función a relation in which each element of the domain is paired with exactly one element in the range range p. P4 rango the set of all ycoordinates of the ordered pairs of a relation
474  Chapter 8  Exponential and Logarithmic Functions and Relations
{(3,1), (0, 2), (2, 4)} Range Domain 3 0 2
1 2 4
Graphing Exponential Functions Then
Now
Why?
You graphed polynomial functions.
1 2
Have you ever received an email that tells you to forward it to 5 friends? If each of those 5 friends then forwards it to 5 of their friends, who each forward it to 5 of their friends, the number of people receiving the email is growing exponentially.
(Lesson 64)
Graph exponential growth functions. Graph exponential decay functions.
The equation y = 5 x can be used to represent this situation, where x is the number of rounds that the email has been forwarded.
NewVocabulary exponential function exponential growth asymptote growth factor exponential decay decay factor
1
Exponential Growth A function like y = 5 x, where the base is a constant and the
exponent is the independent variable, is an exponential function. One type of exponential function is exponential growth. An exponential growth function is a function of the form f(x) = b x, where b > 1. The graph of an exponential function has an asymptote, which is a line that the graph of the function approaches.
KeyConcept Parent Function of Exponential Growth Functions Parent Functions: f (x ) = b x, b > 1 Tennessee Curriculum Standards ✔ 3103.3.11 Describe and articulate the characteristics and parameters of a parent function. SPI 3103.3.5 Describe the domain and range of functions and articulate restrictions imposed either by the operations or by the contextual situations which the functions represent. SPI 3103.3.10 Identify and/ or graph a variety of functions and their translations. Also addresses ✓3103.3.2.
Type of graph:
continuous, onetoone, and increasing
Domain:
all real numbers
Range:
all positive real numbers
Asymptote:
xaxis
Intercept:
(0, 1)
f (x) = b x, b >1
(−1, b1 )
(1, b) (0, 1)
Example 1 Graph Exponential Growth Functions Graph y = 3 x. State the domain and range. Make a table of values. Then plot the points and sketch the graph. x y = 3x x y = 3x
3
2
1 3 3 = _ 27
1 3 2 = _ 9
1
_3
31 = 3
3 2 = √ 27
2 _3
1 _
0
2
3
1 _ 2
√ 3 = _
3
30 = 1
2 32 = 9
14 12 10 8 6 4 2 −4−3−2−10
y
1 2 3 4x
The domain is all real numbers, and the range is all positive real numbers.
GuidedPractice 1. Graph y = 4 x. State the domain and range.
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The graph of f(x) = b x represents a parent graph of the exponential functions. The same techniques used to transform the graphs of other functions you have studied can be applied to the graphs of exponential functions.
KeyConcept Transformations of Exponential Functions f (x ) = ab x  h + k
StudyTip Look Back To review transformations of parent functions, see Lesson 27.
k – Vertical Translation
h – Horizontal Translation ⎪h⎥ units right if h is positive
⎪k⎥ units up if k is positive
⎪h⎥ units left if h is negative
⎪k⎥ units down if k is negative
a – Orientation and Shape
If a < 0, the graph is reflected in the xaxis.
StudyTip End Behavior Remember that end behavior is the action of the graph as x approaches positive infinity or negative infinity. In Example 2a, as x approaches infinity, y approaches infinity. In Example 2b, as x approaches infinity, y approaches negative infinity.
If ⎪a⎥ > 1, the graph is stretched vertically. If 0 < ⎪a⎥ < 1, the graph is compressed vertically.
Example 2 Graph Transformations Graph each function. State the domain and range. a. y = 2 x + 1 The equation represents a translation of the graph of y = 2 x one unit up. x
y = 2x + 1
3 2 1 0 1 2 3
2 3 + 1 = 1.125 2 2 + 1 = 1.25 2 1 + 1 = 1.5 20 + 1 = 2 21 + 1 = 3 22 + 1 = 5 23 + 1 = 9
y
0
x
Domain = {all real numbers}; Range = {y  y > 1}
_
b. y =  1 · 5 x  2 2
The equation represents a transformation of the graph of y = 5 x. Graph y = 5 x and transform the graph. 1 • a = _ : The graph is reflected in the
y
2
y = 5x
xaxis and compressed vertically. • h = 2: The graph is translated 2 units right. • k = 0: The graph is not translated vertically.
y = − 1 · 5x − 2
Domain = {all real numbers}
2
Range = {y  y < 0}
GuidedPractice 2A. y = 2 x + 3  5
476  Lesson 81  Graphing Exponential Functions
0
2B. y = 0.1(6) x  3
x
You can model exponential growth with a constant percent increase over specific time periods using the following function. A(t) = a(1 + r) t The function can be used to find the amount A(t) after t time periods, where a is the initial amount and r is the percent of increase per time period. Note that the base of the exponential expression, 1 + r, is called the growth factor. The exponential growth function is often used to model population growth.
RealWorld Example 3 Graph Exponential Growth Functions CENSUS The first U.S. Census was conducted in 1790. At that time, the population was 3,929,214. Since then, the U.S. population has grown by approximately 2.03% annually. Draw a graph showing the population growth of the U.S. since 1790.
RealWorldLink The U.S. Census Bureau’s American Community Survey is mailed to approximately 1 out of every 480 households.
First, write an equation using a = 3,929,214, and r = 0.0203. y = 3,929,214(1.0203) t Then graph the equation.
Source: Census Bureau
[0, 250] scl: 25 by [0, 400,000,000] scl: 40,000,000
StudyTip Interest The formula for simple interest, i = prt, illustrates linear growth over time. However, the formula for compound interest, A(t ) = a(1 + r ) t, illustrates exponential growth over time. This is why investments with compound interest make more money.
GuidedPractice 3. FINANCIAL LITERACY Teen spending is expected to grow 3.5% annually from $79.7 billion in 2006. Draw a graph to show the spending growth.
2
Exponential Decay The second type of exponential function is exponential decay.
KeyConcept Parent Function of Exponential Decay Functions Parent Functions: f (x ) = b x, 0 < b < 1 Type of graph:
continuous, onetoone, and decreasing
Domain:
all real numbers
Range:
positive real numbers
Asymptote:
xaxis
Intercept:
(0, 1)
Model f(x)
f (x) = b x, 0
(−1, b1 ) (0, 1) 0
(1, b) x
The graphs of exponential decay functions can be transformed in the same manner as those of exponential growth. connectED.mcgrawhill.com
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StudyTip Exponential Decay Be sure not to confuse a dilation in which ⎪a⎥ < 1 with exponential decay in which 0 < b < 1.
v Example 4 Graph Exponential Decay Functions Graph each function. State the domain and range. 1 a. y = _
(3)
x
1 y= _
(3)
x
x
3
(_13 ) = 27 (_13 ) = 9 _ (_13 ) = √3 (_13 ) = 1 (_13 ) = _13 _ 1 (_13 ) = √_ 27 (_13 ) = _19
3
2
2
14 12 10 8 6 4
y
1
1 _
2
2
−4−3−2−10
1 2 3 4x
0
0
1
1
3 2
_3 2
2
2
The domain is all real numbers, and the range is all positive real numbers.
(_4 )
b. y = 2 1
x+2
3 x
1 The equation represents a transformation of the graph of y = _ .
(4)
Examine each parameter. • a = 2: The graph is stretched vertically. • h = 2: The graph is translated 2 units left. • k = 3: The graph is translated 3 units down. y
( 4 )x
y= 1
x
0
( 1 )x + 2 3
y=2 4
The domain is all real numbers, and the range is all real numbers greater than 3.
GuidedPractice 2 x4 4A. y = 3 _ +2
(5)
3 _ 5 4B. y = _
8(6)
x1
+1
Similar to exponential growth, you can model exponential decay with a constant percent of decrease over specific time periods using the following function. A(t) = a(1  r) t The base of the exponential expression, 1  r, is called the decay factor.
478  Lesson 81  Graphing Exponential Functions
RealWorld Example 5 Graph Exponential Decay Functions TEA A cup of green tea contains 35 milligrams of caffeine. The average teen can eliminate approximately 12.5% of the caffeine from their system per hour. a. Draw a graph to represent the amount of caffeine remaining after drinking a cup of green tea. y = a(1  r) t = 35(1  0.125) t = 35(0.875) t Graph the equation.
RealWorldLink After water, tea is the most consumed beverage in the U.S. It can be found in over 80% of American households. Just over half the American population drinks tea daily. Source: The Tea Association of the USA
[0, 10] scl: 1 by [0, 50] scl: 5
b. Estimate the amount of caffeine in a teenager’s body 3 hours after drinking a cup of green tea. y = 35(0.875) t Equation from part a = 35(0.875) 3 Replace t with 3. ≈ 23.45 Use a calculator. The caffeine in a teenager will be about 23.45 milligrams after 3 hours.
GuidedPractice 5. A cup of black tea contains about 68 milligrams of caffeine. Draw a graph to represent the amount of caffeine remaining in the body of an average teen after drinking a cup of black tea. Estimate the amount of caffeine in the body 2 hours after drinking a cup of black tea.
Check Your Understanding
= StepbyStep Solutions begin on page R20.
Examples 1–2 Graph each function. State the domain and range.
Example 3
Example 4
1. f(x) = 2 x
2. f(x) = 5 x
3 f(x) = 3 x  2 + 4
4. f(x) = 2 x + 1 + 3
5. f(x) = 0.25(4) x  6
6. f(x) = 3(2) x + 8
7. SCIENCE A virus spreads through a network of computers such that each minute, 25% more computers are infected. If the virus began at only one computer, graph the function for the first hour of the spread of the virus. Graph each function. State the domain and range. 2 8. f(x) = 2 _
x3
(3) 4 1 _ 10. f(x) = _ 3(5) Example 5
4
x4
+3
12. FINANCIAL LITERACY A new SUV depreciates in value each year by a factor of 15%. Draw a graph of the SUV’s value for the first 20 years after the initial purchase.
3 1 _ 9. f(x) = _
2(4) 1 _ 1 11. f(x) = _ 8(4)
x+1
x+6
+5
+7
All New
Only $20,000
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Practice and Problem Solving
Extra Practice begins on page 947.
Examples 1–2 Graph each function. State the domain and range. 13. f(x) = 2(3) x
14. f(x) = 2(4) x
15. f(x) = 4 x + 1  5
16. f(x) = 3 2x + 1
17. f(x) = 0.4(3) x + 2 + 4
18. f(x) = 1.5(2) x + 6
Example 3
19 SCIENCE The population of a colony of beetles grows 30% each week for 10 weeks. If the initial population is 65 beetles, graph the function that represents the situation.
Example 4
Graph each function. State the domain and range. 3 20. f(x) = 4 _
(5) 3 _ 2 23. f(x) = _ 4(3)
x+4
x+4
Example 5
2 21. f(x) = 3 _
x3
(5) 3 1 _ 24. f(x) = _ 2(8)
+3
2
1 _ 1 22. f(x) = _
x+2
x+5
2(5) 5 _ 4 25. f(x) = _ 4(5)
6 +9
+8
x+4
+2
26. ATTENDANCE The attendance for a basketball team declined at a rate of 5% per game throughout a losing season. Graph the function modeling the attendance if 15 home B games were played and 23,500 people were at the first game. 27. PHONES The function P(x) = 2.28(0.9 x) can be used to model the number of pay phones in millions x years since 1999. a. Graph the function. b. Explain what the P(x)intercept and the asymptote represent in this situation. 28. HEALTH Each day, 10% of a certain drug dissipates from system. a. Graph the function representing this situation. b. How much of the original amount remains in the system after 9 days? c. If a second dose should not be taken if more than 50% of the original amount is in the system, when should the label say it is safe to redose? Design the label and explain your reasoning. 29. NUMBER THEORY A sequence of numbers follows a pattern in which the next number is 125% of the previous number. The first number in the pattern is 18. a. Write the function that represents the situation. b. Graph the function for the first 10 numbers. c. What is the value of the tenth number? Round to the nearest whole number.
C
For each graph, f(x) is the parent function and g(x) is a transformation of f(x). Use the graph to determine the equation of g(x). 30. f(x) = 3 x
31. f(x) = 2 x
32. f(x) = 4 x g(x)
g(x)
g(x)
0
0
x
480  Lesson 81  Graphing Exponential Functions
0
x
x
33
MULTIPLE REPRESENTATIONS In this problem, you will use the tables below for exponential functions f(x), g(x), and h(x). x
1
0
1
2
3
4
5
f (x )
2.5
2
1
1
5
13
29
x
1
0
1
2
3
4
5
5
11
23
47
95
191
383
1
0
1
2
3
4
5
3
2.5
2.25
2.125
2.0625
2.0313
2.0156
g(x ) x h(x )
a. Graphical Graph the functions for 1 ≤ x ≤ 5 on separate graphs. b. Verbal List any function with a negative coefficient. Explain your reasoning. c. Analytical List any function with a graph that is translated to the left. d. Analytical Determine which functions are growth models and which are decay models.
H.O.T. Problems
Use HigherOrder Thinking Skills
34. REASONING Determine whether each statement is sometimes, always, or never true. Explain your reasoning. a. An exponential function of the form y = ab x  h + k has a yintercept. b. An exponential function of the form y = ab x  h + k has an xintercept. x
c. The function f(x) = ⎪b⎥ is an exponential growth function if b is an integer. 3 2 _ 35. ERROR ANALYSIS Vince and Grady were asked to graph f(x) = _ 3 4 them correct? Explain your reasoning.
()
Vince 8 6 4 2 −8−6−4−20 −4 −6 −8
f(x)
2 4 6 8x
x1
. Is either of
Grady 8 6 4 2
f(x)
2 4 6 8x
−8−6−4−20 −4 −6 −8
36. CHALLENGE A substance decays 35% each day. After 8 days, there are 8 milligrams of the substance remaining. How many milligrams were there initially? 8 37. OPEN ENDED Give an example of a value of b for which f(x) = _ b exponential decay.
()
38.
x
represents
E
WRITING IN MATH Write the procedure for transforming the graph of g(x) = b x to the graph of f(x) = ab x  h + k. Justify each step. connectED.mcgrawhill.com
481
SPI 3108.4.3, SPI 3102.3.5
Standardized Test Practice −− −−− 39. GRIDDED RESPONSE In the figure, PO RN, ON = 12, MN = 6, and RN = 4. What is the −− length of PO? 1
41. One hundred students will attend the fall dance if tickets cost $30 each. For each $5 increase in price, 10 fewer students will attend. What price will deliver the maximum dollar sales? F G H J
3 .
0
/
40. Ivan has enough money to buy 12 used CDs. If the cost of each CD was $0.20 less, Ivan could buy 2 more CDs. How much money does Ivan have to spend on CDs? A $16.80 B $16.40
$30 $35 $40 $45
42. SAT/ACT Javier mows a lawn in 2 hours. Tonya mows the same lawn in 1.5 hours. About how many minutes will it take to mow the lawn if Javier and Tonya work together? A 28 minutes B 42 minutes C 51 minutes
C $15.80 D $15.40
D 1.2 hours E 1.4 hours
Spiral Review Solve each equation or inequality. (Lesson 77) y + 5 = √ 2y  3 √ 46. 6 + √ 3y + 4 < 6 43.
44.
y + 1 + √ y4=5 √
45. 10  √ 2x + 7 ≤ 3
47. √ d + 3 + √ d+7>4
48. √ 2x + 5  √ 9+x>0
xy 50. _ 3
3x + 4x 2 51. _ _2
Simplify. (Lesson 76) 1 49. _ _2
√ z
y5
x
3
1 _
4
√ 27 53. _ 4
6 52. √ 27x 3

a 2 54. _ _1 _1
√ 3
6a 3 · a

4
55. FOOTBALL The path of a football thrown across a field is given by the equation y = 0.005x 2 + x + 5, where x represents the distance, in feet, the ball has traveled horizontally and y represents the height, in feet, of the ball above ground level. About how far has the ball traveled horizontally when it returns to ground level? (Lesson 56) 56. COMMUNITY SERVICE A drug awareness program is being presented at a theater that seats 300 people. Proceeds will be donated to a local drug information center. If every two adults must bring at least one student, what is the maximum amount of money that can be raised? (Lesson 34)
Skills Review Simplify. Assume that no variable equals 0. (Lesson 61) 57. f 7 · f 4
58. (3x 2) 3
482  Lesson 81  Graphing Exponential Functions
59. (2y)(4xy 3)
3 2 _ 60. _ c f 4 cd
( 5 )( 3 )
2
Graphing Technology Lab
Solving Exponential Equations and Inequalities You can use a TI83/84 Plus graphing calculator to solve exponential equations by graphing or by using the table feature. To do this, you will write the equations as systems of equations.
Activity 1
_
Solve 3x  4 = 1 . 9
Step 1 Graph each side of the equation as a separate function. Enter 3x  4 as Y1. Be sure to include parentheses around the exponent. 1 Enter _ as Y2. Then graph the two equations. 9
[10, 10] scl: 1 by [1, 1] scl: 0.1
Step 2 Use the intersect feature. You can use the intersect feature on the CALC menu to approximate the ordered pair of the point at which the graphs cross. The calculator screen shows that the xcoordinate of the point at which the curves cross is 2. Therefore, the solution of the equation is 2. [10, 10] scl: 1 by [1, 1] scl: 0.1
Step 3 Use the TABLE feature. You can also use the table feature to locate the point at which the curves intersect. The table displays xvalues and corresponding yvalues for each graph. Examine the table to find the xvalue for which the yvalues of the graphs are equal. − 1 . Thus, the solution At x = 2, both functions have a yvalue of 0.1 or _ 9
of the equation is 2. CHECK
Substitute 2 for x in the original equation. 1 3x  4 _ Original equation 9
32  4
1 _
9 _ 2 3 1 9 _1 = _1 ✓ 9 9
Substitute 2 for x. Simplify. The solution checks.
A similar procedure can be used to solve exponential inequalities. (continued on the next page) connectED.mcgrawhill.com
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Graphing Technology Lab
Solving Exponential Equations and Inequalities Continued Activity 2 Description Solve 2x  2 ≥ 0.5x  3. Step 1 Enter the related inequalities. Rewrite the problem as a system of inequalities. The first inequality is 2x  2 ≥ y or y ≤ 2x  2. Since this inequality includes the less than or equal to symbol, shade below the curve. First enter the boundary, and then use the arrow and to choose the shade below icon,
keys
.
0.5x  3.
The second inequality is y ≥ Shade above the curve since this inequality contains greater than or equal to. 2
KEYSTROKES:
2
.5 3
Step 2 Graph the system. KEYSTROKES:
The xvalues of the points in the region where the shadings overlap is the solution set of the original inequality. Using the intersect feature, you can conclude that the solution set is {x  x ≥ 2.5}. [10, 10] scl: 1 by [10, 10] scl: 1
Step 3 Use the TABLE feature. Verify using the TABLE feature. Set up the table to show xvalues in increments of 0.5. KEYSTROKES:
[TBLSET] 0
.5
[TABLE]
Notice that for xvalues greater than x = 2.5, Y1 > Y2. This confirms that the solution of the inequality is {x  x ≥ 2.5}.
Exercises Solve each equation or inequality. 1 1. 9x  1 = _
2. 4x + 3 = 25x
3. 5x  1 = 2x
4. 3.5x + 2 = 1.75x + 3
5. 3x + 4 = 0.52x + 3
6. 62  x  4 < 0.25x  2.5
7. 16x  1 > 22x + 2
8. 3x  4 ≤ 5 2
81
_x
9. 5x + 3 ≤ 2x + 4
10. WRITING IN MATH Explain why this technique of graphing a system of equations or inequalities works to solve exponential equations and inequalities.
484  Explore 82  Graphing Technology Lab: Solving Exponential Equations and Inequalities
Solving Exponential Equations and Inequalities Then
Now
Why?
You graphed exponential functions.
1 2
Membership on Internet social networking sites tends to increase exponentially. The membership growth of one Web site can be modeled by the equation y = 2.2(1.37) x, where x is the number of years since 2004 and y is the number of members in millions.
(Lesson 81)
Solve exponential equations. Solve exponential inequalities.
You can use y = 2.2(1.37) x to determine how many members there will be in a given year, or to determine the year in which membership was at a certain level.
NewVocabulary exponential equation compound interest exponential inequality
1
Solve Exponential Equations In an exponential equation, variables occur as exponents.
KeyConcept Property of Equality for Exponential Functions Words
Let b > 0 and b ≠ 1. Then b x = b y if and only if x = y.
Example
If 3 x = 3 5, then x = 5. If x = 5, then 3 x = 3 5.
Tennessee Curriculum Standards SPI 3103.3.13 Solve contextual problems using quadratic, rational, radical and exponential equations, finite geometric series or systems of equations.
The Property of Equality can be used to solve exponential equations.
Example 1 Solve Exponential Equations Solve each equation. a. 2 x = 8 3 2x = 83 2 x = (2 3) 3 2x = 29 x=9
Original equation Rewrite 8 as 2 3. Power of a Power Property of Equality for Exponential Functions
b. 9 2x  1 = 3 6x 9 2x  1 = 3 6x (3 2) 2x  1 = 3 6x 3 4x  2 = 3 6x 4x  2 = 6x 2 = 2x 1 = x
Original equation Rewrite 9 as 3 2. Power of a Power Property of Equality for Exponential Functions Subtract 4x from each side. Divide each side by 2.
GuidedPractice 1A. 4 2n  1 = 64
1B. 5 5x = 125 x + 2
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You can use information about growth or decay to write the equation of an exponential function.
RealWorld Example 2 Write an Exponential Function SCIENCE Kristin starts an experiment with 7500 bacteria cells. After 4 hours, there are 23,000 cells. a. Write an exponential function that could be used to model the number of bacteria after x hours if the number of bacteria changes at the same rate. At the beginning of the experiment, the time is 0 hours and there are 7500 bacteria cells. Thus, the yintercept, and the value of a, is 7500. When x = 4, the number of bacteria cells is 23,000. Substitute these values into an exponential function to determine the value of b. y = ab x 23,000 = 7500 · b 4 3.067 ≈ b 4 4 √ 3.067 ≈ b 1.323 ≈ b
Exponential function Replace x with 4, y with 23,000, and a with 7500. Divide each side by 7500. Take the 4th root of each side. Use a calculator.
An equation that models the number of bacteria is y ≈ 7500(1.323) x. b. How many bacteria cells can be expected in the sample after 12 hours? y ≈ 7500(1.323) x
Modeling equation
≈ 7500(1.323) 12
Replace x with 12.
≈ 215,665
Use a calculator.
There will be approximately 215,665 bacteria cells after 12 hours.
GuidedPractice 2. RECYCLING A manufacturer distributed 3.2 million aluminum cans in 2005. A. In 2010, the manufacturer distributed 420,000 cans made from the recycled cans it had previously distributed. Assuming that the recycling rate continues, write an equation to model the distribution each year of cans that are made from recycled aluminum. B. How many cans made from recycled aluminum can be expected in the year 2050?
RealWorldLink In 2008, the U.S. recycling rate for metals of 35% prevented the release of approximately 25 million metric tons of carbon into the air—roughly the amount emitted annually by 4.5 million cars. Source: Environmental Protection Agency
Exponential functions are used in situations involving compound interest. Compound interest is interest paid on the principal of an investment and any previously earned interest.
KeyConcept Compound Interest You can calculate compound interest using the following formula. A = P 1 + _r ,
(
n
)nt
where A is the amount in the account after t years, P is the principal amount invested, r is the annual interest rate, and n is the number of compounding periods each year.
486  Lesson 82  Solving Exponential Equations and Inequalities
Example 3 Compound Interest An investment account pays 4.2% annual interest compounded monthly. If $2500 is invested in this account, what will be the balance after 15 years? Understand Find the total amount in the account after 15 years. Plan Use the compound interest formula. P = 2500, r = 0.042, n = 12, and t = 15 r Solve A = P 1 + _
(
n
)nt
Compound Interest Formula
0.042 = 2500 1 + _
(
12
)
12 · 15
≈ 4688.87
P = 2500, r = 0.042, n = 12, t = 15 Use a calculator.
Check Graph the corresponding equation y = 2500(1.0035) 12t. Use CALC: value to find y when x = 15. The yvalue 4688.8662 is very close to 4688.87, so the answer is reasonable. [0, 20] scl: 1 by [0, 10,000] scl: 1000
WatchOut!
GuidedPractice
Percents Remember to convert all percents to decimal form; 4.2% is 0.042.
3. Find the account balance after 20 years if $100 is placed in an account that pays 1.2% interest compounded twice a month.
2
Solve Exponential Inequalities An exponential inequality is an inequality involving exponential functions.
KeyConcept Property of Inequality for Exponential Functions Words
Let b > 1. Then b x > b y if and only if x > y, and b x < b y if and only if x < y.
Example
If 2 x > 2 6, then x > 6. If x > 6, then 2 x > 2 6. This property also holds true for ≤ and ≥.
Example 4 Solve Exponential Inequalities Solve 16 2x  3 < 8. 16 2x  3 < 8
Original inequality
(2 4) 2x  3 < 2 3 2 8x  12 < 2 3 8x  12 < 3
Rewrite 16 as 2 4 and 8 as 2 3. Power of a Power Property of Inequality for Exponential Functions
8x < 15 15 x<_ 8
Add 12 to each side. Divide each side by 8.
GuidedPractice Solve each inequality. 1 4A. 3 2x  1 ≥ _ 243
1 4B. 2 x + 2 > _ 32
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Check Your Understanding Example 1
Example 2
= StepbyStep Solutions begin on page R20.
Solve each equation. 1. 3 5x = 27 2x  4
2. 16 2y  3 = 4 y + 1
3. 2 6x = 32 x  2
4. 49 x + 5 = 7 8x  6
5. SCIENCE Mitosis is a process in which one cell divides into two. The Escherichia coli is one of the fastest growing bacteria. It can reproduce itself in 15 minutes. a. Write an exponential function to represent the number of cells c after t minutes. b. If you begin with one Escherichia coli cell, how many cells will there be in one hour?
Example 3
Example 4
6. A certificate of deposit (CD) pays 2.25% annual interest compounded biweekly. If you deposit $500 into this CD, what will the balance be after 6 years? Solve each inequality. 1 8. 25 y  3 ≤ _
( 125 )
7. 4 2x + 6 ≤ 64 2x  4
y+2
Practice and Problem Solving Example 1
Solve each equation. 9. 8 4x + 2 = 64
Example 2
Extra Practice begins on page 947.
10. 5 x  6 = 125
11 81 a + 2 = 3 3a + 1
12. 256 b + 2 = 4 2  2b
13. 9 3c + 1 = 27 3c  1
14. 8 2y + 4 = 16 y + 1
15. MONEY In 2009, MyLien received $10,000 from her grandmother. Her parents invested all of the money, and by 2021, the amount will have grown to $16,960. a. Write an exponential function that could be used to model the money y. Write the function in terms of x, the number of years since 2009. b. Assume that the amount of money continues to grow at the same rate. What would be the balance in the account in 2031? Write an exponential function for the graph that passes through the given points.
Example 3
16. (0, 6.4) and (3, 100)
17. (0, 256) and (4, 81)
18. (0, 128) and (5, 371,293)
19. (0, 144), and (4, 21,609)
20. Find the balance of an account after 7 years if $700 is deposited into an account paying 4.3% interest compounded monthly. 21. Determine how much is in a retirement account after 20 years if $5000 was invested at 6.05% interest compounded weekly. 22. A savings account offers 0.7% interest compounded bimonthly. If $110 is deposited in this account, what will the balance be after 15 years? 23. A college savings account pays 13.2% annual interest compounded semiannually. What is the balance of an account after 12 years if $21,000 was initially deposited?
Example 4
Solve each inequality. 24. 625 ≥ 5 a + 8 1 26. _
c2
< 32 2c
3t + 5
1 ≥ _
( 64 ) 1 28. (_ 9)
( 243 )
25. 10 5b + 2 > 1000 1 27. _ t6
( 27 ) 1 29. (_ 36 )
488  Lesson 82  Solving Exponential Equations and Inequalities
2d  2
≤ 81 d + 4
w+2
1 < _
( 216 )
4w
B
30. SCIENCE A mug of hot chocolate is 90°C at time t = 0. It is surrounded by air at a constant temperature of 20°C. If stirred steadily, its temperature in Celsius after t minutes will be y(t) = 20 + 70(1.071) t. a. Find the temperature of the hot chocolate after 15 minutes. b. Find the temperature of the hot chocolate after 30 minutes. c. The optimum drinking temperature is 60°C. Will the mug of hot chocolate be at or below this temperature after 10 minutes? 31 ANIMALS Studies show that an animal will defend a territory, with area in square yards, that is directly proportional to the 1.31 power of the animal’s weight in pounds. a. If a 45pound beaver will defend 170 square yards, write an equation for the area a defended by a beaver weighing w pounds. b. Scientists believe that thousands of years ago, the beaver’s ancestors were 11 feet long and weighed 430 pounds. Use your equation to determine the area defended by these animals. Solve each equation. 1 32. _
(2) 1 35. (_ 8)
4x + 1
= 8 2x + 1
3x + 4
1 = _
(4)
2x + 4
1 33. _
(5) 2 36. (_ 3)
x5
= 25 3x + 2
5x + 1
27 = _
(8)
x4
1 34. 216 = _
(6)
25 37. _
( 81 )
2x + 1
x+3
729 = _
( 125 )
3x + 1
38. POPULATION In 1950, the world population was about 2.556 billion. By 1980, it had increased to about 4.458 billion. a. Write an exponential function of the form y = abx that could be used to model the world population y in billions for 1950 to 1980. Write the equation in terms of x, the number of years since 1950. (Round the value of b to the nearest tenthousandth.) b. Suppose the population continued to grow at that rate. Estimate the population in 2000. c. In 2000, the population of the world was about 6.08 billion. Compare your estimate to the actual population. d. Use the equation you wrote in Part a to estimate the world population in the year 2020. How accurate do you think the estimate is? Explain your reasoning.
C
39. TREES The diameter of the base of a tree trunk in centimeters varies directly with 3 the _ power of its height in meters. 2 a. A young sequoia tree is 6 meters tall, and the diameter of its base is 19.1 centimeters. Use this information to write an equation for the diameter d of the base of a sequoia tree if its height is h meters high. b. Refer to the information at the left. Find the diameter of the General Sherman Tree at its base. 40. FINANCIAL LITERACY Mrs. Jackson Option B: Option A: has two different retirement 6.5% annual rate 4.2% annual rate investment plans from which compounded monthly; compounded quarterly; minimum deposit to choose. minimum deposit $5000 $5000 a. Write equations for Option A PLUS 2.3% annual rate and Option B given the compounded weekly; minimum deposits. minimum deposit $5000 b. Draw a graph to show the balances for each investment option after t years. c. Explain whether Option A or Option B is the better investment choice. connectED.mcgrawhill.com
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41.
MULTIPLE REPRESENTATIONS In this problem, you will explore the rapid increase of an exponential function. A large sheet of paper is cut in half, and one of the resulting pieces is placed on top of the other. Then the pieces in the stack are cut in half and placed on top of each other. Suppose this procedure is repeated several times. a. Concrete Perform this activity and count the number of sheets in the stack after the first cut. How many pieces will there be after the second cut? How many pieces after the third cut? How many pieces after the fourth cut? b. Tabular Record your results in a table. c. Symbolic Use the pattern in the table to write an equation for the number of pieces in the stack after x cuts. d. Analytical The thickness of ordinary paper is about 0.003 inch. Write an equation for the thickness of the stack of paper after x cuts. e. Analytical How thick will the stack of paper be after 30 cuts?
H.O.T. Problems
Use HigherOrder Thinking Skills
42. WRITING IN MATH In a problem about compound interest, describe what happens as the compounding period becomes more frequent while the principal and overall time remain the same. 43. ERROR ANALYSIS Beth and Liz are solving 6 x  3 > 36 x  1. Is either of them correct? Explain your reasoning.
Liz
Beth x–3
–x – 1
6 > 36 x–3 6 > (6 2) –x – 1 6 x – 3 > 6 –2x – 2 x – 3 > –2x – 2 3x > 1 1 x>_ 3
x–3
6 > 36 –x – 1 6 x – 3 > (6 2) –x – 1 6 x – 3 > 6 –x + 1 x – 3 > –x + 1 2x > 4 x>2
44. CHALLENGE Solve for x: 16 18 + 16 18 + 16 18 + 16 18 + 16 18 = 4 x. 45. OPEN ENDED What would be a more beneficial change to a 5year loan at 8% interest compounded monthly: reducing the term to 4 years or reducing the interest rate to 6.5%? 46. REASONING Determine whether the following statements are sometimes, always, or never true. Explain your reasoning. a. 2 x > 8 20x for all values of x. b. The graph of an exponential growth equation is increasing. c. The graph of an exponential decay equation is increasing. 47. OPEN ENDED Write an exponential inequality with a solution of x ≤ 2. 48. PROOF Show that 27 2x · 81 x + 1 = 3 2x + 2 · 9 4x + 1. 49. WRITING IN MATH If you were given the initial and final amounts of a radioactive substance and the amount of time that passes, how would you determine the rate at which the amount was increasing or decreasing in order to write an equation?
490  Lesson 82  Solving Exponential Equations and Inequalities
SPI 3102.3.5, SPI 3102.1.3
Standardized Test Practice 50. 3 × 10 4 = A 0.003 B 0.0003
52. GRIDDED RESPONSE The three angles of a triangle are 3x, x + 10, and 2x  40. Find the measure of the smallest angle in the triangle.
C 0.00003 D 0.000003
53. SAT/ACT Which of the following is equivalent to (x)(x)(x)(x) for all x?
51. Which of the following could not be a solution to 5  3x < 3? F 2.5 G 3
D 4x 2 E x4
A x+4 B 4x C 2x 2
H 3.5 J 4
Spiral Review Graph each function. (Lesson 81) x
55. y = 5(2) x
1 56. y = 4 _
57. √ x+53=0
58. √ 3t  5  3 = 4
4 59. √ 2x  1 = 2
60. √ x6
3 61. √5m +2=3
62. (6n  5) 3 + 3 = 2
54. y = 2(3) x
(3)
Solve each equation. (Lesson 77)
√ x
=3
_1
_1
_1
63. (5x + 7) 5 + 3 = 5
64. (3x  2) 5 + 6 = 5
_1
65. (7x  1) 3 + 4 = 2
66. SALES A salesperson earns $10 an hour plus a 10% commission on sales. Write a function to describe the salesperson’s income. If the salesperson wants to earn $1000 in a 40hour week, what should his sales be? (Lesson 72) 67. STATE FAIR A dairy makes three types of cheese—cheddar, Monterey Jack, and Swiss—and sells the cheese in three booths at the state fair. At the beginning of one day, the first booth received x pounds of each type of cheese. The second booth received y pounds of each type of cheese, and the third booth received z pounds of each type of cheese. By the end of the day, the dairy had sold 131 pounds of cheddar, 291 pounds of Monterey Jack, and 232 pounds of Swiss. The table below shows the percent of the cheese delivered in the morning that was sold at each booth. How many pounds of cheddar cheese did each booth receive in the morning? (Lesson 35) Booth 1
Booth 2
Booth 3
Cheddar
Type
40%
30%
10%
Monterey Jack
40%
90%
80%
Swiss
30%
70%
70%
Skills Review Find [g ◦ h](x) and [h ◦ g](x). (Lesson 71) 68. h(x) = 2x  1 g(x) = 3x + 4
69. h(x) = x 2 + 2 g(x) = x  3
70. h(x) = x 2 + 1 g(x) = 2x + 1
71. h(x) = 5x g(x) = 3x  5
72. h(x) = x 3 g(x) = x  2
73. h(x) = x + 4 g(x) = ⎪x⎥ connectED.mcgrawhill.com
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Logarithms and Logarithmic Functions Then
Now
Why?
You found the inverse of a function.
1 2
Many scientists believe the extinction of the dinosaurs was caused by an asteroid striking Earth. Astronomers use the Palermo scale to classify objects near Earth based on the likelihood of impact. To make comparing several objects easier, the scale was developed using logarithms. The Palermo scale value of any object can be found using the equation PS = log 10 R, where R is the relative risk posed by the object.
(Lesson 72)
NewVocabulary logarithm logarithmic function
Evaluate logarithmic expressions. Graph logarithmic functions.
1
Logarithmic Functions and Expressions Consider the exponential function
f(x) = 2 x and its inverse. Recall that you can graph an inverse function by interchanging the x and yvalues in the ordered pairs of the function. y = 2x
Tennessee Curriculum Standards ✔ 3103.1.8 Understand and describe the inverse relationship between exponential and logarithmic functions. ✔ 3103.3.17 Know that the logarithm and exponential functions are inverses and use this information to solve realworld problems. SPI 3103.3.10 Identify and/ or graph a variety of functions and their translations.
y
x = 2y
x
y
x
y
3
_1
_1
8
8
3
2
_1
_1
4
4
1
_1
_1
2
2
0 1 2 3
1 2 4 8
1 2 4 8
(2, 4)
2
(0, 1)
1
O
The inverse of y = 2 x can be defined as x = 2 y. In general, the inverse of y = b x is x = b y. In x = b y, the variable y is called the logarithm of x. This is usually written as y = log b x, which is read y equals log base b of x.
y = 2x y=x y (4, 2) x = 2
(1, 0)
x
As the value of y decreases, the value of x approaches 0.
0 1 2 3
KeyConcept Logarithm with Base b Words
Let b and x be positive numbers, b ≠ 1. The logarithm of x with base b is denoted log b x and is defined as the exponent y that makes the equation b y = x true.
Symbols
Suppose b > 0 and b ≠ 1. For x > 0, there is a number y such that log b x = y if and only if b y = x.
Example
If log 3 27 = y, then 3 y = 27.
The definition of logarithms can be used to express logarithms in exponential form.
Example 1 Logarithmic to Exponential Form Write each equation in exponential form. a. log 2 8 = 3 log 2 8 = 3 → 8 = 2 3
_
b. log 4 1 = 4 256
1 1 log 4 _ = 4 → _ = 4 4 256
GuidedPractice 1A. log 4 16 = 2
492  Lesson 83
1B. log 3 729 = 6
256
The definition of logarithms can also be used to write exponential equations in logarithmic form.
Example 2 Exponential to Logarithmic Form
StudyTip
Write each equation in logarithmic form.
Continuity Most exponential and logarithmic functions are continuous. In Example 2a, f (x) is increasing from 0 to infinity.
a. 15 3 = 3375
_1
b. 4 2 = 2 _1
1 4 2 = 2 → log 4 2 = _
15 3 = 3375 → log 15 3375 = 3
2
GuidedPractice
_1
2A. 4 3 = 64
2B. 125 3 = 5
You can use the definition of a logarithm to evaluate a logarithmic expression.
WatchOut! Logarithmic Base It is easy to get confused about which number is the base and which is the exponent in logarithmic equations. Consider highlighting each number as you solve to help organize your calculations.
Example 3 Evaluate Logarithmic Expressions Evaluate log 16 4. log 16 4 = y 4 = 16 y
Let the logarithm equal y. Definition of logarithm
2 y
1
4 = (4 ) 4 1 = 4 2y 1 = 2y
16 = 4 2 Power of a Power Property of Equality for Exponential Functions
_1 = y
Divide each side by 2.
2
1 Thus, log 16 4 = _ . 2
GuidedPractice Evaluate each expression. 3A. log 3 81
2
3B. log _1 256 2
Graphing Logarithmic Functions The function y = log b x, where b ≠ 1, is called
a logarithmic function. The graph of f(x) = log b x represents a parent graph of the logarithmic functions.
KeyConcept Parent Function of Logarithmic Functions Parent function:
f (x ) = log b x
Type of graph:
continuous, onetoone
Domain:
all positive real numbers
Range:
all real numbers
Asymptote:
f (x )axis
Intercept:
(1, 0)
f(x)
(1, 0) 0
( 1b , −1)
f (x) = log b x, b >1
f(x)
(b, 1)
f (x) = log b x, 0
(b, 1) x
0
(1, 0)
x
( 1b , −1)
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Example 4 Graph Logarithmic Functions Graph each function. a. f(x) = log 5 x Step 1 Identify the base. b=5
StudyTip Zero Exponent Recall that for any b ≠ 0, b 0 = 1. Therefore, log 2 0 is undefined because 2 x ≠ 0 for any xvalue.
Step 2 Determine points on the graph. 1 Because 5 > 1, use the points _ , 1 , (1, 0), b and (b, 1).
(
f(x)
)
(1, 0) 0
Step 3 Plot the points and sketch the graph.
(5, 1) x
( 15 , −1)
(_b1, 1) → (_15 , 1) (1, 0) (b, 1) → (5, 1)
b. f(x) = log _1 x 3
1 Step 1 b = _
f(x)
3
( 13 , 1)
1 Step 2 0 < _ < 1, so use the points 3
0
_1 , 1 , (1, 0) and (3, 1). 3
( )
(1, 0)
(3, −1)
Step 3 Sketch the graph.
GuidedPractice 4A. f(x) = log 2 x
4B. f(x) = log _1 x 8
The same techniques used to transform the graphs of other functions you have studied can be applied to the graphs of logarithmic functions.
KeyConcept Transformations of Logarithmic Functions f ((x ) = a llog b ((x  h) h +k h – Horizontal Translation
k – Vertical Translation
h units right if h is positive ⎪h⎥ units left if h is negative
k units up if k is positive ⎪k⎥ units down if k is negative a – Orientation and Shape
If a < 0, the graph is reflected across the xaxis.
494  Lesson 83  Logarithms and Logarithmic Functions
If ⎪a⎥ > 1, the graph is stretched vertically. If 0 < ⎪a⎥ < 1, the graph is compressed vertically.
x
Example 5 Graph Logarithmic Functions
StudyTip
Graph each function.
End Behavior In Example 5a, as x approaches infinity, f (x ) approaches infinity.
a. f(x) = 3 log 10 x + 1
f(x)
This represents a transformation of the graph of f(x) = log 10 x. • ⎪a⎥ = 3: The graph stretches vertically
x
0
• h = 0: There is no horizontal shift. • k = 1: The graph is translated 1 unit up.
_
b. f(x) = 1 log _1 (x  3) 2
f(x)
4
This is a transformation of the graph of f(x) = log _1 x.
x
4
•
⎪a⎥
1 =_ : The graph is compressed vertically.
0
2
• h = 3: The graph is translated 3 units to the right. • k = 0: There is no vertical shift.
GuidedPractice Graph each function. 1 5B. f(x) = _ log _1 (x + 1)  5
5A. f(x) = 2 log 3 (x  2)
4
2
RealWorld Example 6 Find Inverses of Exponential Functions EARTHQUAKES The Richter scale measures earthquake intensity. The increase in intensity between each number is 10 times. For example, an earthquake with a rating of 7 is 10 times more intense than one measuring 6. The intensity of an earthquake can be modeled by y = 10 x  1, where x is the Richter scale rating. a. Use the information at the left to find the intensity of the strongest recorded earthquake in the United States.
RealWorldLink The largest recorded earthquake in the United States was a magnitude 9.2 that struck Prince William Sound, Alaska, on Good Friday, March 28, 1964. Source: United States Geological Survey
y = 10 x  1 = 10 9.2  1 = 10 8.2 = 158,489,319.2
Original equation Substitute 9.2 for x. Simplify. Use a calculator.
b. Write an equation of the form y = log 10 x + c for the inverse of the function. y = 10 x  1 x = 10 y  1 y  1 = log 10 x y = log 10 x + 1
Original equation Replace x with y, replace y with x, and solve for y. Definition of logarithm Add 1 to each side.
GuidedPractice 6. Write an equation for the inverse of the function y = 0.5 x.
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495
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R20.
Write each equation in exponential form. 1. log 8 512 = 3
Example 2
2. log 5 625 = 4
Write each equation in logarithmic form.
_3
3. 11 3 = 1331 Example 3
4. 16 4 = 8
Evaluate each expression. 1 6. log 2 _
5. log 13 169
7. log 6 1
128
Examples 4–5 Graph each function. 8. f(x) = log 3 x
9. f(x) = log _1 x 6
10. f(x) = 4 log 4 (x  6) Example 6
11. f(x) = 2 log _ 1 x5 10
12. SCIENCE Use the information at the beginning of the lesson. The Palermo scale value of any object can be found using the equation PS = log 10 R, where R is the relative risk posed by the object. Write an equation in exponential form for the inverse of the function.
Practice and Problem Solving Example 1
Example 2
Write each equation in exponential form. 13. log 2 16 = 4
14. log 7 343 = 3
1 15. log 9 _ = 2
1 16. log 3 _ = 3 27
17. log 12 144 = 2
18. log 9 1 = 0
81
Write each equation in logarithmic form. 1 19. 9 1 = _
1 20. 6 3 = _
22. 4 6 = 4096
23. 27 3 = 9
9
Example 3
Extra Practice begins on page 947.
_2
21. 2 8 = 256
216
_3
24. 25 2 = 125
Evaluate each expression. 1 25. log 3 _
1 26. log 4 _
27. log 8 512
28. log 6 216
29. log 27 3
30. log 32 2
31. log 9 3
32. log 121 11
33 log _1 3125
34. log _1 512
1 35. log _1 _
1 36. log _1 _
9
64
5
8
3
81
6
216
Examples 4–5 Graph each function. 37. f(x) = log 6 x
38. f(x) = log _1 x
39. f(x) = 4 log 2 x + 6
40. f(x) = log _1 x
41. f(x) = log 10 x
42. f(x) = 3 log _ 1 x+2
43. f(x) = 6 log _1 (x + 2)
44. f(x) = 8 log 3 (x  4)
45. f(x) = log _1 (x + 1)  9
46. f(x) = log 5 (x  4)  5
1 1 47. f(x) = _ log 8 (x  3) + 4 48. f(x) = _ log _1 (x + 2)  5
9
8
5
496  Lesson 83  Logarithms and Logarithmic Functions
6
12
4
3
6
Example 6
1 49. PHOTOGRAPHY The formula n = log 2 _ represents the change in the fstop setting n to p
use in less light where p is the fraction of sunlight. a. Benito’s camera is set up to take pictures in direct sunlight, but it is a cloudy day. If 1 the amount of sunlight on a cloudy day is _ as bright as direct sunlight, how many 4
fstop settings should he move to accommodate less light? b. Graph the function. c. Use the graph in part b to predict what fraction of daylight Benito is accommodating if he moves down 3 fstop settings. Is he allowing more or less light into the camera?
50. EDUCATION To measure a student’s retention of knowledge, the student is tested after a given amount of time. A student’s score on an Algebra 2 test t months after the school year is over can be approximated by y(t) = 85  6 log 2 (t + 1), where y(t) is the student’s score as a percent. a. What was the student’s score at the time the school year ended (t = 0)? b. What was the student’s score after 3 months? c. What was the student’s score after 15 months?
Graph each function. 51 f(x) = 4 log 2 (2x  4) + 6
52. f(x) = 3 log 12 (4x + 3) + 2
53. f(x) = 15 log 14 (x + 1)  9
54. f(x) = 10 log 5 (x  4)  5
1 55. f(x) = _ log 8 (x  3) + 4
1 56. f(x) = _ log 6 (6x + 2)  5
6
B
3
57. ADVERTISING In general, the more money a company spends on advertising, the higher the sales. The amount of money in sales for a company, in thousands, can be modeled by the equation S(a) = 10 + 20 log 4(a + 1), where a is the amount of money spent on advertising in thousands, when a ≥ 0. a. The value of S(0) ≈ 10, which means that if $10 is spent on advertising, $10,000 is returned in sales. Find the values of S(3), S(15), and S(63). b. Interpret the meaning of each function value in the context of the problem. c. Graph the function. d. Use the graph in part c and your answers from part a to explain why the money spent in advertising becomes less “efficient” as it is used in larger amounts.
58. BIOLOGY The generation time for bacteria is the time that it takes for the population to double. The generation time G for a specific type of bacteria can be found using t experimental data and the formula G = _ , where t is the time period, b is the 3.3 log b f
number of bacteria at the beginning of the experiment, and f is the number of bacteria at the end of the experiment. a. The generation time for mycobacterium tuberculosis is 16 hours. How long will it take four of these bacteria to multiply into 1024 bacteria? b. An experiment involving rats that had been exposed to salmonella showed that the generation time for the salmonella was 5 hours. After how long would 20 of these bacteria multiply into 8000? c. E. coli are fast growing bacteria. If 6 E. coli can grow to 1296 in 4.4 hours, what is the generation time of E. coli? connectED.mcgrawhill.com
497
C
59 FINANCIAL LITERACY Jacy has spent $2000 on a credit card. The credit card company charges 24% interest, compounded monthly. The credit card company uses A _ log = 12t to determine how much time it will be until Jacy’s debt 0.24 _
(1+
12
) 2000
reaches a certain amount, if A is the amount of debt after a period of time, and t is time in years. a. Graph the function for Jacy’s debt. b. Approximately how long will it take Jacy’s debt to double? c. Approximately how long will it be until Jacy’s debt triples?
H.O.T. Problems
Use HigherOrder Thinking Skills
60. WHICH ONE DOESN’T BELONG? Find the expression that does not belong. Explain.
log4 16
log 2 16
log 2 4
log 3 9
61. CHALLENGE Consider y = log b x in which b, x, and y are real numbers. Zero can be in the domain sometimes, always or never. Justify your answer. 62. ERROR ANALYSIS Betsy says that the graphs of all logarithmic functions cross the yaxis at (0, 1) because any number to the zero power equals 1. Tyrone disagrees. Is either of them correct? Explain your reasoning. 63. REASONING Without using a calculator, compare log7 51, log 8 61, and log 9 71. Which of these is the greatest? Explain your reasoning. 64. OPEN ENDED Write a logarithmic equation of the form y = logb x for each of the following conditions. a. y is equal to 25.
b. y is negative.
c. y is between 0 and 1.
d. x is 1.
e. x is 0. 65. ERROR ANALYSIS Elisa and Matthew are evaluating log _1 49. Is either of them correct? 7 Explain your reasoning.
Elisa
Matthew
log _1 49 = y
log _1 49 = y
7
_1 y = 49 7
(7 –1) y = 7 2 (7) –y = 7 2 y=2
7
49 y = _17 y (7 2) = (7) –1 7 2y = (7) –1 2y = –1 y = – _12
66. WRITING IN MATH A transformation of log 10 x is g(x) = a log 10 (x  h) + k. Explain the process of graphing this transformation.
498  Lesson 83  Logarithms and Logarithmic Functions
SPI 3102.3.5, SPI 3108.4.3, SPI 3102.3.9
Standardized Test Practice 67. A rectangle is twice as long as it is wide. If the width of the rectangle is 3 inches, what is the area of the rectangle in square inches? A 9 B 12 C 15 D 18 68. SAT/ACT Ichiro has some pizza. He sold 40% more slices than he ate. If he sold 70 slices of pizza, how many did he eat? F 25 J 98 G 50 K 100 H 75
69. SHORT RESPONSE In the figure AB = BC, CD = BD, and m∠CAD = 70°. What is the measure of angle ADC? $
#
"
%
70. If 6x  3y = 30 and 4x = 2  y then find x + y. A 4 B 2 C 2 D 4
Spiral Review Solve each inequality. Check your solution. (Lesson 82) 71. 3 n  2 > 27
1 72. 2 2n ≤ _ 16
73. 16 n < 8 n + 1
74. 32 5p + 2 ≥ 16 5p
77. y = 30 x
78. y = 0.2(5) x
Graph each function. (Lesson 81) 1 75. y =  _
x
(5)
76. y = 2.5(5) x
79. GEOMETRY The area of a triangle with sides of length a, b, and c is given by 1 s(s  a)(s  b)(s  c) , where s = _ (a + b + c). If the lengths of the sides of √ 2 a triangle are 6, 9, and 12 feet, what is the area of the triangle expressed in radical form? (Lesson 75)
12
6
9
80. GEOMETRY The volume of a rectangular box can be written as 6x 3 + 31x 2 + 53x + 30 when the height is x + 2. (Lesson 65) a. What are the width and length of the box? b. Will the ratio of the dimensions of the box always be the same regardless of the value of x? Explain. 81. AUTO MECHANICS Shandra is inventory manager for a local repair shop. She orders 6 batteries, 5 cases of spark plugs, and two dozen pairs of wiper blades and pays $830. She orders 3 batteries, 7 cases of spark plugs, and four dozen pairs of wiper blades and pays $820. The batteries are $22 less than twice the price of a dozen wiper blades. Use augmented matrices to determine what the cost of each item on her order is. (Lesson 46)
Skills Review Solve each equation or inequality. Check your solution. (Lesson 82) 1 82. 9 x = _ 81
83. 2 6x = 4 5x + 2
84. 49 3p + 1 = 7 2p  5
2
85. 9 x ≤ 27 x
22
connectED.mcgrawhill.com
499
Graphing Technology Lab
Choosing the Best Model We can find exponential and logarithmic functions of best fit using a TI83/84 Plus graphing calculator.
Tennessee Curriculum Standards SPI 3103.1.3 Use technology tools to identify and describe patterns in data using nonlinear and transcendental functions that approximate data as well as using those functions to solve contextual problems. Also addresses ✓3103.1.1, ✓3103.1.3, ✓3103.1.5, ✓3103.1.10, SPI 3103.1.3, CLE 3103.5.1, ✓3103.5.1, ✓3103.5.6, ✓3103.5.7, SPI 3103.5.3, SPI 3103.5.6, and SPI 3103.5.7.
Activity The population per square mile in the United States has changed dramatically over a period of years. The table shows the number of people per square mile for several years. a. Use a graphing calculator to enter the data. Then draw a scatter plot that shows how the number of people per square mile is related to the year. Step 1 Enter the year into L1 and the people per square mile into L2. KEYSTROKES:
See pages 94 and 95 to review how to enter lists.
Be sure to clear the Y= list. Use the key to move the cursor from L1 to L2.
U.S. Population Density Year 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890
People per square mile 4.5 6.1 4.3 5.5 7.4 9.8 7.9 10.6 10.9 14.2 17.8
Year 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
People per square mile 21.5 26.0 29.9 34.7 37.2 42.6 50.6 57.5 64.0 70.3 80.0
Source: NortheastMidwest Institute
Step 2 Draw the scatter plot. KEYSTROKES:
See pages 94 and 95 to review how to graph a scatter plot.
Make sure that Plot 1 is on, the scatter plot is chosen, Xlist is L1, and Ylist is L2.
[1780, 2020] scl: 10 by [0, 115] scl: 5
Step 3 Find a regression equation. To find an equation that best fits the data, use the regression feature of the calculator. Examine various regressions to determine the best model. Recall that the calculator returns the correlation coefficient r, which is used to indicate how well the model fits the data. The closer r is to 1 or 1, the better the fit. Linear regression 4
KEYSTROKES:
KEYSTROKES:
Quadratic regression 5 r2 = 0.9974003374 r = √ 0.9974003374 r ≈ 0.9986993228
500  Extend 83  Graphing Technology Lab: Choosing the Best Model
Exponential regression 0
KEYSTROKES:
Power regression KEYSTROKES:
Compare the rvalues. Linear: 0.945411996 Exponential: 0.991887235
[A]
Quadratic: 0.9986993228 Power: 0.9917543535
The rvalue of the quadratic regression is closest to 1, so it appears to best model the data. You can examine the equation visually by graphing the regression equation with the scatter plot. 5
5
KEYSTROKES:
1
b. If this trend continues, what will be the population per square mile in 2020?
[1780, 2020] scl: 10 by [0, 115] scl: 5
To determine the population per square mile in 2020, find the value of y when x = 2020. KEYSTROKES:
[CALC]
2020
If this trend continues, there will be approximately 94.9 people per square mile.
[1780, 2020] scl: 10 by [0, 115] scl: 5
Exercises For Exercises 1–5, Jewel deposited $50 into an account, then forgot about it and made no further deposits or withdrawals. The table shows the account balance for several years.
Elapsed Time (years)
Balance
0
$50.00
1. Use a graphing calculator to draw a scatter plot of the data.
2
$55.80
2. Calculate and graph a curve of fit using an exponential regression.
4
$64.80
3. Write the equation of best fit.
6
$83.09
8
$101.40
10
$123.14
12
$162.67
4. Based on the model, what will the account balance be after 25 years? 5. Is an exponential model the best fit for the data? Explain. 6. YOUR TURN Write a question that can be answered by examining the data of a logarithmic model. First choose a topic and then collect relevant data through Internet research or a survey. Next, make a scatter plot and find a regression equation for your data. Then answer your question.
connectED.mcgrawhill.com
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Solving Logarithmic Equations and Inequalities Then
Now
Why?
You evaluated logarithmic expressions.
1 2
Each year the National Weather Service documents about 1000 tornado touchdowns in the United States. The intensity of a tornado is measured on the Fujita scale. Tornados are divided into six categories according to their wind speed, path length, path width, and damage caused.
(Lesson 83)
NewVocabulary logarithmic equation logarithmic inequality
Solve logarithmic equations. Solve logarithmic inequalities.
1
FScale
Wind Speed (mph)
Type of Damage
F0 Gale
4072
chimneys, branches
F1 Moderate
73112
mobile homes overturned
F2 Significant
113157
roof torn off
F3 Severe
158206
tree uprooted
F4 Devastating
207260
homes leveled, cars thrown
F5 Incredible
261318
homes thrown
F6 Inconceivable
319379
level has never been achieved
Solve Logarithmic Equations A logarithmic equation contains one or
more logarithms. You can use the definition of a logarithm to help you solve logarithmic equations.
Example 1 Solve a Logarithmic Equation Tennessee Curriculum Standards ✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
3 Solve log 36 x = _ . 2
3 log 36 x = _ 2
Original equation _3
x = 36 2
Definition of logarithm _3
x = (6 2) 2
36 = 6 2
x = 6 3 or 216
Power of a Power
GuidedPractice Solve each equation. 3 1A. log 9 x = _ 2
5 1B. log 16 x = _ 2
Use the following property to solve logarithmic equations that have logarithms with the same base on each side.
KeyConcept Property of Equality for Logarithmic Functions
502  Lesson 84
Symbols
If b is a positive number other than 1, then log b x = log b y if and only if x = y.
Example
If log 5 x = log 5 8, then x = 8. If x = 8, then log 5 x = log 5 8.
SPI 2.14
Test Example 2 Standardized Test Example 2 Solve log 2 (x 2  4) = log 2 3x.
StudyTip
A 2
Substitution To save time, you can substitute each answer choice in the original equation to find the one that results in a true statement.
Read the Test Item
B 1
C 2
D 4
You need to find x for the logarithmic equation. Solve the Test Item log 2 (x 2  4) = log 2 3x
Original equation
2
x  4 = 3x x 2  3x  4 = 0 (x  4)(x + 1) = 0 x  4 = 0 or x + 1 = 0 x=4 x = 1
Property of Equality for Logarithmic Functions Subtract 3x from each side. Factor. Zero Product Property Solve each equation.
CHECK Substitute each value into the original equation. x=4 x = 1 2 log 2 (4  4) log 2 3(4) log 2 [(1) 2  4] log 2 3(1) log 2 12 = log 2 12 log 2 (3) log 2 (3) The domain of a logarithmic function cannot be 0, so log 2 (3) is undefined and 1 is an extraneous solution. The answer is D.
GuidedPractice 2. Solve log 3 (x 2  15) = log 3 2x. F 3
2
G 1
H 5
J
15
Solve Logarithmic Inequalities A logarithmic inequality is an inequality that
involves logarithms. The following property can be used to solve logarithmic inequalities.
KeyConcept Property of Inequality for Logarithmic Functions If b > 1, x > 0, and log b x > y, then x > b y.
Math HistoryLink Zhang Heng (A.D. 78–139) The earliest known seismograph was invented by Zhang Heng in China in 132 B.C. It was a large brass vessel with a heavy pendulum and several arms that tripped when an earthquake tremor was felt. This helped determine the direction of the quake.
If b > 1, x > 0, and log b x < y, then 0 < x < b y. This property also holds true for ≤ and ≥.
Example 3 Solve a Logarithmic Inequality Solve log 3 x > 4. log 3 x > 4 x > 34 x > 81
Original inequality Property of Inequality for Logarithmic Functions Simplify.
GuidedPractice Solve each inequality. 3A. log 4 x ≥ 3
3B. log 2 x < 4 connectED.mcgrawhill.com
503
The following property can be used to solve logarithmic inequalities that have logarithms with the same base on each side. Exclude from your solution set values that would result in taking the logarithm of a number less than or equal to zero in the original inequality.
KeyConcept Property of Inequality for Logarithmic Functions Symbols
If b > 1, then log b x > log b y if and only if x > y, and log b x < log b y if and only if x < y.
Example
If log 6 x > log 6 35, then x > 35 This property also holds true for ≤ and ≥.
Example 4 Solve Inequalities with Logarithms on Each Side Solve log 4 (x + 3) > log 4 (2x + 1). log 4 (x + 3) > log 4 (2x + 1)
Original inequality
x + 3 > 2x + 1
Property of Inequality for Logarithmic Functions
2>x
Subtract x + 1 from each side.
1 Exclude all values of x for which x + 3 ≤ 0 or 2x + 1 ≤ 0. So, x > 3, x > _ , and 2 ⎧ ⎫ 1 1 _ _ x < 2. The solution set is ⎨x  < x < 2⎬ or  , 2 . 2 2 ⎩ ⎭
(

)
GuidedPractice 4. Solve log 5 (2x + 1) ≤ log 5 (x + 4). Check your solution.
Check Your Understanding Example 1
= StepbyStep Solutions begin on page R20.
Solve each equation. 4 1. log 8 x = _
3 2. log 16 x = _
3
Example 2
3. MULTIPLE CHOICE Solve log 5 (x 2  10) = log 5 3x. A 10
Example 3
4
B 2
C 5
D 2, 5
Solve each inequality. 4. log 5 x > 3
5. log 8 x ≤ 2
6. log 4 (2x + 5) ≤ log 4 (4x  3)
7. log 8 (2x) > log 8 (6x  8)
Practice and Problem Solving
Extra Practice begins on page 947.
Examples 1–2 Solve each equation. 3 8. log 81 x = _ 4
1 11. log 6 _ =x 36
5 9. log 25 x = _
1 10. log 8 _ =x
2 5 12. log x 32 = _ 2
2
3 13. log x 27 = _ 2
14. log 3 (3x + 8) = log 3 (x 2 + x)
15 log 12 (x 2  7) = log 12 (x + 5)
16. log 6 (x 2  6x) = log 6 (8)
17. log 9 (x 2  4x) = log 9 (3x  10)
18. log 4 (2x 2 + 1) = log 4 (10x  7)
19. log 7 (x 2  4) = log 7 (x + 2)
504  Lesson 84  Solving Logarithmic Equations and Inequalities
SCIENCE The equation for wind speed w, in miles per hour, near the center of a tornado is w = 93 log 10 d + 65, where d is the distance in miles that the tornado travels. 20. Write this equation in exponential form. 21. In May of 1999, a tornado devastated Oklahoma City with the fastest wind speed ever recorded. If the tornado traveled 525 miles, estimate the wind speed near the center of the tornado. Solve each inequality. Examples 3–4 22. log 6 x < 3
B
23. log 4 x ≥ 4
24. log 3 x ≥ 4
25 log 2 x ≤ 2
26. log 5 x > 2
27. log 7 x < 1
28. log 2 (4x  6) > log 2 (2x + 8)
29. log 7 (x + 2) ≥ log 7 (6x  3)
30. log 3 (7x  6) < log 3 (4x + 9)
31. log 5 (12x + 5) ≤ log 5 (8x + 9)
32. log 11 (3x  24) ≥ log 11 (5x  8)
33. log 9 (9x + 4) ≤ log 9 (11x  12)
34. SCIENCE The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The magnitude M is given by M = log 10 x, where x represents the amplitude of the seismic wave causing ground motion. a. How many times as great is the amplitude caused by an earthquake with a Richter scale rating of 8 as an aftershock with a Richter scale rating of 5? b. In 1906, San Francisco was almost completely destroyed by a 7.8 magnitude earthquake. In 1911, an earthquake estimated at magnitude 8.1 occurred along the New Madrid fault in the Mississippi River Valley. How many times greater was the New Madrid earthquake than the San Francisco earthquake? 35. MUSIC The first key on a piano keyboard corresponds to a pitch with a frequency of 27.5 cycles per second. With every successive key, going up the black and white keys, the pitch multiplies by a constant. The formula for the frequency of the pitch sounded f 27.5
when the nth note up the keyboard is played is given by n = 1 + 12 log 2 _. a. A note has a frequency of 220 cycles per second. How many notes up the piano keyboard is this? b. Another pitch on the keyboard has a frequency of 880 cycles per second. After how many notes up the keyboard will this be found? 36.
MULTIPLE REPRESENTATIONS In this problem, you will explore the graphs shown: y = log 4 x and y = log _1 x. 4
a. Analytical How do the shapes of the graphs compare? How do the asymptotes and the xintercepts of the graphs compare? b. Verbal Describe the relationship between the graphs.
[2, 8] scl: 1 by [5, 5] scl: 1
c. Graphical Use what you know about transformations of graphs to compare and contrast the graph of each function and the graph of y = log 4 x. 1. y = log 4 x + 2 2. y = log 4 (x + 2) 3. y = 3 log 4 x d. Analytical Describe the relationship between y = log 4 x and y = 1(log 4 x). What are a reasonable domain and range for each function? e. Analytical Write an equation for a function for which the graph is the graph of y = log 3 x translated 4 units left and 1 unit up. connectED.mcgrawhill.com
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C
37 SOUND The relationship between the intensity of sound I and the number of I decibels β is β = 10 log 10 _ , where I is the intensity of sound in watts per 10 12 square meter.
)
(
a. Find the number of decibels of a sound with an intensity of 1 watt per square meter. b. Find the number of decibels of sound with an intensity of 10 2 watts per square meter. c. The intensity of the sound of 1 watt per square meter is 100 times as much as the intensity of 10 2 watts per square meter. Why are the decibels of sound not 100 times as great?
H.O.T. Problems
Sound
Intensity
Decibels
pin drop normal breathing clothes dryer subway train ﬁrecracker
100 101
0 1
106 1010 1012
6 10 12
Use HigherOrder Thinking Skills
38. ERROR ANALYSIS Ryan and Heather are solving log 3 x ≥ 3. Is either of them correct? Explain your reasoning.
Ryan log 3 x ≥ –3
Heather log3 x ≥ –3
x ≥ 3 –3
x ≥ 3–3
x≥_ 27
0 < x ≤_ 27
1
1
39. CHALLENGE Find log 3 27 + log 9 27 + log 27 27 + log 81 27 + log 243 27. 40. REASONING The Property of Inequality for Logarithmic Functions states that when b > 1, log b x > log b y if and only if x > y. What is the case for when 0 < b < 1? Explain your reasoning. 41. WRITING IN MATH Explain how the domain and range of logarithmic functions are related to the domain and range of exponential functions. 42. OPEN ENDED Give an example of a logarithmic equation that has no solution. 43. REASONING Choose the appropriate term. Explain your reasoning. All logarithmic equations are of the form y = log b x. a. If the base of a logarithmic equation is greater than 1 and the value of x is between 0 and 1, then the value for y is (less than, greater than, equal to) 0. b. If the base of a logarithmic equation is between 0 and 1 and the value of x is greater than 1, then the value of y is (less than, greater than, equal to) 0. c. There is/are (no, one, infinitely many) solution(s) for b in the equation y = log b 0. d. There is/are (no, one, infinitely many) solution(s) for b in the equation y = log b 1. 44. WRITING IN MATH Explain why any logarithmic function of the form y = log b x has an xintercept of (1, 0) and no yintercept.
506  Lesson 84  Solving Logarithmic Equations and Inequalities
SPI 3103.5.1, SPI 3103.5.8
Standardized Test Practice 6.4 4 45. Find x if _ =_ . x
A B C D
47. Clara received a 10% raise each year for 3 consecutive years. What was her salary after the three raises if her starting salary was $12,000 per year?
7
3.4 9.4 11.2 44.8
46. The monthly precipitation in Houston for part of a year is shown. Month
Precipitation (in.)
April May June July August
3.60 5.15 5.35 3.18 3.83
A B C D
$14,520 $15,972 $16,248 $16,410
48. SAT/ACT A vendor has 14 helium balloons for sale: 9 are yellow, 3 are red, and 2 are green. A balloon is selected at random and sold. If the balloon sold is yellow, what is the probability that the next balloon, selected at random, is also yellow? 36 H _
1 F _
Find the median precipitation. F 4.25 in. H 3.83 in. G 4.22 in. J 3.60 in.
91 _ J 8 13
9
1 G _ 8
9 K _ 14
Spiral Review Evaluate each expression. (Lesson 83) 49. log 4 256
1 50. log 2 _
51. log 6 216
52. log 3 27
1 53. log 5 _
54. log 7 2401
8
125
Solve each equation or inequality. Check your solution. (Lesson 82) 55. 5 2x + 3 ≤ 125
56. 3 3x  2 > 81
57. 4 4a + 6 ≤ 16 a
58. 11 2x + 1 = 121 3x
59. 3 4x  7 = 27 2x + 3
60. 8 x  4 ≤ 2 4  x
61. SHIPPING The height of a shipping cylinder is 4 feet more than the radius. If the volume of the cylinder is 5π cubic feet, how tall is it? Use the formula V = π · r 2 · h. (Lesson 68) 62. NUMBER THEORY Two complex conjugate numbers have a sum of 12 and a product of 40. Find the two numbers. (Lesson 54)
Skills Review Simplify. Assume that no variable equals zero. (Lesson 61) 63. x 5 · x 3
64. a 2 · a 6
66. (3b 3c 2) 2
x y 67. _ 2
4 6
xy
65. (2p 2n) 3
(d )
c9 68. _ 7
0
connectED.mcgrawhill.com
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MidChapter Quiz
Tennessee Curriculum Standards
Lessons 81 through 84
SPI 3103.1.1, SPI 2.14
Graph each function. State the domain and range. (Lesson 81) 1. f(x ) = 3(4)
x
11. MULTIPLE CHOICE Which graph below is the graph of the function f (x ) = log 3 (x + 5) + 3? (Lesson 83) f(x)
A
2. f(x ) = (2) x + 5
C
3. f(x ) = 0.5(3) x + 2 + 4 2 4. f(x ) = 3 _
x1
(3)
+8 O
B
b. How many bacteria cells can be expected after 4 hours?
8 6 4 2
f(x)
O
2 4 6 8 10 12 14 16 x
−4 −6 −8
D
2 4 6 8 10 12 14 16 x
−4 −6 −8
6. MULTIPLE CHOICE Which exponential function has a graph that passes through the points at (0, 125) and (3, 1000)?
f(x)
O
x
5. SCIENCE You are studying a bacteria population. The population originally started with 6000 bacteria cells. After 2 hours, there were 28,000 bacteria cells. (Lesson 81) a. Write an exponential function that could be used to model the number of bacteria after x hours if the number of bacteria changes at the same rate.
8 6 4 2
8 6 4 2 O
f(x)
2 4 6 8 10 12 14 x
−4 −6 −8
(Lesson 81)
A f(x ) = 125(3) x B f(x ) = 1000(3) x C f(x ) = 125(1000) x D f(x ) = 125(2)
x
Evaluate each expression. (Lesson 83) 12. log 4 32 13. log 5 5 12 14. log 16 4
7. POPULATION In 1995, a certain city had a population of 45,000. It increased to 68,000 by 2007. (Lesson 82) a. What is an exponential function that could be used to model the population of this city x years after 1995? b. Use your model to estimate the population in 2020. 8. MULTIPLE CHOICE Find the value of x for log 3 (x 2 + 2x ) = log 3 (x + 2). (Lesson 83) F x = 2, 1
15. Write log 9 729 = 3 in exponential form. (Lesson 83)
Solve each equation or inequality. Check your solution. (Lessons 82 and 84)
16. 3 x = 27 2 17. 4 3x  1 = 16 x 1 18. _ = 243 2x + 1 9
19. 16 2x + 3 < 64
G x = 2 x+3
H x=1
1 20. _
J no solution
3 21. log 4 x = _
( 32 )
≥ 16 3x 2
Graph each function. (Lesson 83) 9. f(x ) = 3 log 2 (x  1) 10. f(x ) = 4 log 3 (x  2) + 5
508  Chapter 8  MidChapter Quiz
22. log 7 (x + 3) = log 7 (6x + 5) 23. log 2 x < 3 24. log 8 (3x + 7) = log 8 (2x  5)
Properties of Logarithms Product
Then You evaluated logarithmic expressions and solved logarithmic equations. (Lesson 84)
Tennessee Curriculum Standards ✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
Now
1 2
Why?
Simplify and evaluate expressions using the properties of logarithms. Solve logarithmic equations using the properties of logarithms.
1
The level of acidity in food is important to some consumers with sensitive stomachs. Most of the foods that we consume are more acidic than basic. The pH scale measures acidity; a low pH indicates an acidic solution, and a high pH indicates a basic solution. It is another example of a logarithmic scale based on powers of ten. Black coffee has a pH of 5, while neutral water has a pH of 7. Black coffee is one hundred times as acidic as neutral water, because 10 7  5 = 10 2 or 100.
pH Level
Lemon Juice
2.1
Sauerkraut
3.5
Tomatoes
4.2
Black Coffee
5.0
Milk
6.4
Pure Water
7.0
Eggs
7.8
Milk of Magnesia
10.0
Properties of Logarithms Since logarithms are exponents, the properties of
logarithms can be derived from the properties of exponents. The Product Property of Logarithms can be derived from the Product of Powers Property of Exponents.
KeyConcept Product Property of Logarithms Words
The logarithm of a product is the sum of the logarithms of its factors.
Symbols
For all positive numbers a, b, and x, where x ≠ 1, log x ab = log x a + log x b.
Example
log 2 [(5)(6)] = log 2 5 + log 2 6
To show that this property is true, let b x = a and b y = c. Then, using the definition of logarithm, x = log b a and y = log b c. b xb y = ac b x + y = ac log b b x + y = log b ac x + y = log b ac log b a + log b c = log b ac
Substitution Product of Powers Property of Equality for Logarithmic Functions Inverse Property of Exponents and Logarithms Replace x with log b a and y with log b c.
You can use the Product Property of Logarithms to approximate logarithmic expressions.
Example 1 Use the Product Property Use log 4 3 ≈ 0.7925 to approximate the value of log 4 192. log 4 192 = log 4 (4 3 · 3) = log 4 4 3 + log 4 3 = 3 + log 4 3 ≈ 3 + 0.7925 or 3.7925
Replace 192 with 64 · 3 or 4 3 · 3. Product Property Inverse Property of Exponents and Logarithms Replace log 4 3 with 0.7925.
GuidedPractice 1. Use log 4 2 = 0.5 to approximate the value of log 4 32. connectED.mcgrawhill.com
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Recall that the quotient of powers is found by subtracting exponents. The property for the logarithm of a quotient is similar. Let b x = a and b y = c. Then log b a = x and log b c = y bx _ = _a by
c
b x  y = _a
Quotient Property
c
log b b x  y = log b _a
Property of Equality for Logarithmic Equations
c
x  y = log b _a c log b a  log b c = log b _a c
Inverse Property of Exponents and Logarithms Replace x with log b a and y with log b c.
KeyConcept Quotient Property of Logarithms Words
The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.
Symbols
For all positive numbers a, b, and x, where x ≠ 1, log x _a = log x a  log x b.
Example
5 log 2 _ = log 2 5  log 2 6
b
6
RealWorld Example 2 Quotient Property SCIENCE The pH of a substance is defined as the concentration of hydrogen ions 1 [H +] in moles. It is given by the formula pH = log 10 _ . Find the amount of + hydrogen in a liter of acid rain that has a pH of 4.2.
H
Understand The formula for finding pH and the pH of the rain is given. You want to find the amount of hydrogen in a liter of this rain. Plan Write the equation. Then, solve for [H +]. 1 pH = log 10 _ +
Solve
RealWorldLink Acid rain is more acidic than normal rain. Smoke and fumes from burning fossil fuels rise into the atmosphere and combine with the moisture in the air to form acid rain. Acid rain can be responsible for the erosion of statues, as in the photo above.
Original equation
H _ 4.2 = log 10 1+ H
Substitute 4.2 for pH.
4.2 = log 10 1  log 10 H
+
4.2 = 0  log 10 H + 4.2 = log 10 H
+
4.2 = log 10 H + 10
4.2
=H
+
Quotient Property log 10 1 = 0 Simplify. Multiply each side by 1. Definition of logarithm
There are 10 4.2, or about 0.000063, mole of hydrogen in a liter of this rain. 1 Check 4.2 = log 10 _ + H
pH = 4.2
1 4.2 log 10 _ 4.2
H + = 10 4.2
4.2 log 10 1  log 10 10 4.2 4.2 0  (4.2) 4.2 = 4.2
Quotient Property
10
Simplify.
GuidedPractice 2. SOUND The loudness L of a sound, measured in decibels, is given by L = 10 log 10 R, where R is the sound’s relative intensity. Suppose one person talks with a relative intensity of 10 6 or 60 decibels. How much louder would 100 people be, talking at the same intensity?
510  Lesson 85  Properties of Logarithms
Recall that the power of a power is found by multiplying exponents. The property for the logarithm of a power is similar.
KeyConcept Power Property of Logarithms Words
The logarithm of a power is the product of the logarithm and the exponent.
Symbols
For any real number p, and positive numbers m and b, where b ≠ 1, log b m p = p log b m.
Example
log 2 6 5 = 5 log 2 6
StudyTip
Example 3 Power Property of Logarithms
Answer Check You can check this answer by evaluating 2 4.6438 on a calculator. The calculator should give a result of about 25, since log 2 25 ≈ 4.6438
Given log 2 5 ≈ 2.3219, approximate the value of log 2 25.
means 2
4.6438
log 2 25 = log 2 5 2 = 2 log 2 5 ≈ 2(2.3219) or 4.6438
Replace 25 with 5 2. Power Property Replace log 2 5 with 2.3219.
≈ 25.
GuidedPractice 3. Given log 3 7 ≈ 1.7712, approximate the value of log 3 49.
2
Solve Logarithmic Equations You can use the properties of logarithms to solve equations involving logarithms.
Example 4 Solve Equations Using Properties of Logarithms Solve log 6 x + log 6 (x  9) = 2. log 6 x + log 6 (x  9) = 2
Original equation
log 6 x (x  9) = 2 x(x  9) = 6
Product Property 2
2
x  9x  36 = 0 (x  12)(x + 3) = 0 x  12 = 0
x+3=0
or
x = 12
x = 3
Definition of logarithm Subtract 36 from each side. Factor. Zero Product Property Solve each equation.
CHECK log 6 x + log 6 (x  9) = 2
log 6 x + log 6 (x  9) = 2
log 6 12 + log 6 (12  9) 2
log 6 (3) + log 6 (3  9) 2
log 6 12 + log 6 3 2
log 6 (3) + log 6 (12) 2
log 6 (12 · 3) 2 log 6 36 2
Because log 6 (3) and log 6 (12) are undefined, 3 is an extraneous solution.
2=2 The solution is x = 12.
GuidedPractice 4A. 2 log 7 x = log 7 27 + log 7 3
4B. log 6 x + log 6 (x + 5) = 2 connectED.mcgrawhill.com
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Check Your Understanding
= StepbyStep Solutions begin on page R20.
Examples 1–2 Use log 4 3 ≈ 0.7925 and log 4 5 ≈ 1.1610 to approximate the value of each expression. 1. log 4 18
2. log 4 15
5 3. log 4 _
3 4. log 4 _
3
Example 2
Example 3
4
5. MOUNTAIN CLIMBING As elevation increases, the atmospheric air pressure decreases. The formula for pressure based on elevation is a = 15,500(5  log 10 P), where a is the altitude in meters and P is the pressure in pascals (1 psi ≈ 6900 pascals). What is the air pressure at the summit in pascals for each mountain listed in the table at the right?
Country
Height (m)
Everest Trisuli Bonete McKinley Logan
Nepal/Tibet India Argentina/Chile United States Canada
8850 7074 6872 6194 5959
Given log 3 5 ≈ 1.465 and log 5 7 ≈ 1.2091, approximate the value of each expression. 6. log 3 25
Example 4
Mountain
7. log 5 49
Solve each equation. Check your solutions. 8. log 4 48  log 4 n = log 4 6
9. log 3 2x + log 3 7 = log 3 28
10. 3 log 2 x = log 2 8
11. log 10 a + log 10 (a  6) = 2
Practice and Problem Solving
Extra Practice begins on page 947.
Examples 1–2 Use log 4 2 = 0.5, log 4 3 ≈ 0.7925, and log 4 5 ≈ 1.1610 to approximate the value of each expression.
Example 2
12. log 4 30
13. log 4 20
2 14. log 4 _
4 15 log 4 _ 3
16. log 4 9
17. log 4 8
3
18. SCIENCE In 2007, an earthquake near San Francisco registered approximately 5.6 on the Richter scale. The famous San Francisco earthquake of 1906 measured 8.3 in magnitude. a. How much more intense was the 1906 earthquake than the 2007 earthquake? b. Richter himself classified the 1906 earthquake as having a magnitude of 8.3. More recent research indicates it was most likely a 7.9. What is the difference in intensities?
Example 3
Example 4
Year
Location
Magnitude
1906 1923 1932 1960 1964 2007
San Francisco Tokyo, Japan Gansu, China Chile Alaska San Francisco
8.3 8.3 7.6 9.5 9.2 5.6
Source: TLC
Given log 6 8 ≈ 1.1606 and log 7 9 ≈ 1.1292, approximate the value of each expression. 19. log 6 48
20. log 7 81
21. log 6 512
22. log 7 729
Solve each equation. Check your solutions. 23. log 3 56  log 3 n = log 3 7
24. log 2 (4x) + log 2 5 = log 2 40
25. 5 log 2 x = log 2 32
26. log 10 a + log 10 (a + 21) = 2
512  Lesson 85  Properties of Logarithms
B
27 PROBABILITY In the 1930s, Dr. Frank Benford demonstrated a way to determine whether a set of numbers has been randomly chosen or manually chosen. If the sets 1 of numbers were not randomly chosen, then the Benford formula, P = log 10 1 + _ ,
(
d
)
predicts the probability of a digit d being the first digit of the set. For example, there is a 4.6% probability that the first digit is 9. a. Rewrite the formula to solve for the digit if given the probability. b. Find the digit that has a 9.7% probability of being selected. c. Find the probability that the first digit is 1 (log 10 2 ≈ 0.30103). Use log 5 3 ≈ 0.6826 and log 5 4 ≈ 0.8614 to approximate the value of each expression. 28. log 5 40
29. log 5 30
3 30. log 5 _ 4
4 31. log 5 _
32. log 5 9
33. log 5 16
34. log 5 12
35. log 5 27
3
Solve each equation. Check your solutions. 36. log 3 6 + log 3 x = log 3 12
37. log 4 a + log 4 8 = log 4 24
38. log 10 18  log 10 3x = log 10 2
39. log 7 100  log 7 (y + 5) = log 7 10
1 40. log 2 n = _ log 2 27 + log 2 36 3
1 41. 3 log 10 8  _ log 10 36 = log 10 x 2
Solve for n. 42. log a 6n  3 log a x = log a x
43. 2 log b 16 + 6 log b n = log b (x  2)
Solve each equation. Check your solutions. 44. log 10 z + log 10 (z + 9) = 1
45. log 3 (a 2 + 3) + log 3 3 = 3
46. log 2 (15b  15)  log 2 (b 2 + 1) = 1
47. log 4 (2y + 2)  log 4 (y  2) = 1
48. log 6 0.1 + 2 log 6 x = log 6 2 + log 6 5
8 49. log 7 64  log 7 _ + log 7 2 = log 7 4p 3
50. ENVIRONMENT The humpback whale is an endangered species. Suppose there are 5000 humpback whales in existence today, and the population decreases at a rate of 4% per year. a. Write a logarithmic function for the time in years based upon population. b. After how long will the population drop below 1000? Round your answer to the nearest year. State whether each equation is true or false. 51. log 8 (x  3) = log 8 x  log 8 3
52. log 5 22x = log 5 22 + log 5 x
53. log 10 19k = 19 log 10 k
54. log 2 y 5 = 5 log 2 y
x 55. log 7 _ = log 7 x  log 7 3
56. log 4 (z + 2) = log 4 z + log 4 2
57. log 8 p 4 = (log 8 p) 4
58. log 9 _ = 2 log 9 x + 3 log 9 y  4 log 9 z 4
3
x 2y 3 z
59. PARADE An equation for loudness L, in decibels, is L = 10 log 10 R, where R is the relative intensity of the sound. a. Solve 120 = 10 log 10R to find the relative intensity of the Macy’s Thanksgiving Day Parade with a loudness of 120 decibels depending on how close you are. b. Some parents with young children want the decibel level lowered to 80. How many times less intense would this be? In other words, find the ratio of their intensities. connectED.mcgrawhill.com
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C
60. FINANCIAL LITERACY The average American carries a credit card debt of approximately r b _
(n) $8600 with an annual percentage rate (APR) of 18.3%. The formula m = __ r 1 1+_
(
n
)nt
can be used to compute the monthly payment m that is necessary to pay off a credit card balance b in a given number of years t, where r is the annual percentage rate and n is the number of payments per year. a. What monthly payment should be made in order to pay off the debt in exactly three years? What is the total amount paid?
( mn ) b. The equation t = __ _r can be used to br log 1  _
(
n log 1 +
n
Payment (m )
)
Years (t )
$50
calculate the number of years necessary for a given payment schedule. Copy and complete the table.
$100
c. Graph the information in the table from part b.
$250
$150 $200 $300
d. If you could only afford to pay $100 a month, will you be able to pay off the debt? If so, how long will it take? If not, why not? e. What is the minimum monthly payment that will work toward paying off the debt?
H.O.T. Problems
Use HigherOrder Thinking Skills
61. OPEN ENDED Write a logarithmic expression for each condition. Then write the expanded expression. a. a product and a quotient b. a product and a power c. a product, a quotient, and a power 62. PROOF Use the properties of exponents to prove the Power Property of Logarithms. 63. WRITING IN MATH Explain why the following are true. a. log b 1 = 0 b. log b b = 1
c. log b b x = x
64. CHALLENGE Simplify log √a (a 2) to find an exact numerical value. 65. WHICH ONE DOESN’T BELONG? Find the expression that does not belong. Explain.
logb 24 = logb 2 + logb 12
log b 24 = log b 20 + log b 4
log b 24 = log b 8 + log b 3
log b 24 = log b 4 + log b 6
1 66. REASONING Use the properties of logarithms to prove that log a _ = log a x. x
67. CHALLENGE Simplify x 3 log x 2  log x 5 to find an exact numerical value. 68. WRITING IN MATH Explain how the properties of exponents and logarithms are related. Include examples like the one shown at the beginning of the lesson illustrating the Product Property, but with the Quotient Property and Power Property of Logarithms.
514  Lesson 85  Properties of Logarithms
SPI 3103.5.1, SPI 3103.3.10, SPI 3103.1.1, SPI 3103.3.2
Standardized Test Practice 71. SHORT RESPONSE In y = 6.5(1.07) x, x represents the number of years since 2000, and y represents the approximate number of millions of Americans 7 years of age and older who went camping two or more times that year. Describe how the number of millions of Americans who go camping is changing over time.
69. Find the mode of the data. 22, 11, 12, 23, 7, 6, 17, 15, 21, 19 A 11 B 15
C 16 D There is no mode.
70. SAT/ACT What is the effect on the graph of y = 4x 2 when the equation is changed to y = 2x 2?
72. What are the xintercepts of the graph of y = 4x 2  3x  1?
F The graph is rotated 90 degrees about the origin. G The graph is narrower. H The graph is wider. J The graph of y = 2x 2 is a reflection of the graph y = 4x 2 across the xaxis. K The graph is unchanged.
1 1 A _ and _
C 1 and 1
4 1 _ B 1 and 4 4
1 D 1 and _ 4
Spiral Review Solve each equation. Check your solutions. (Lesson 84) 73. log 5 (3x  1) = log 5 (2x 2)
74. log 10 (x 2 + 1) = 1
75. log 10 (x 2  10x) = log 10 (21)
77. log 4 16 x
78. log 3 27 x
Evaluate each expression. (Lesson 83) 76. log 10 0.001
79. ELECTRICITY The amount of current in amperes I that an appliance uses can be _1
P 2 calculated using the formula I = _ , where P is the power in watts and R is
(R)
the resistance in ohms. How much current does an appliance use if P = 120 watts and R = 3 ohms? Round to the nearest tenth. (Lesson 76) Determine whether each pair of functions are inverse functions. Write yes or no. (Lesson 72) 80. f(x) = x + 73 g(x) = x  73
81. g(x) = 7x  11 1 h(x) = _ x + 11 7
82. SCULPTING Antonio is preparing to make an ice sculpture. He has a block of ice that he wants to reduce in size by shaving off the same amount from the length, width, and height. He wants to reduce the volume of the ice block to 24 cubic feet. (Lesson 67) a. Write a polynomial equation to model this situation.
3 ft
4 ft
5 ft
b. How much should he take from each dimension?
Skills Review Solve each equation or inequality. Check your solution. (Lessons 81 through 84) 83. 3 4x = 3 3  x 86. 49 x = 7 x
2
 15
1 84. 3 2n ≤ _
85. 3 5x · 81 1  x = 9 x  3
87. log 2 (x + 6) > 5
88. log 5 (4x  1) = log 5 (3x + 2)
9
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Common Logarithms Then
Now
Why?
You simplified expressions and solved equations using properties of logarithms.
1
Seismologists use the Richter scale to measure the strength or magnitude of earthquakes. The magnitude of an earthquake is determined using the logarithm of the amplitude of waves recorded by seismographs.
(Lesson 84)
2
Solve exponential equations and inequalities using common logarithms. Evaluate logarithmic expressions using the Change of Base Formula.
Richter Number
The logarithmic scale used by the Richter scale is based on the powers of 10. For example, a magnitude 6.4 earthquake can be represented by 6.4 = log 10 x.
NewVocabulary common logarithm Change of Base Formula
1
Intensity
1
10 1 Micro
2
10 2 Minor
3
10 3 Minor
4
10 4 Light
5
10 5 Moderate
6
10 6 Strong
7
10 7 Major
8
10 8 Great
Common Logarithms You have seen that the base 10 logarithm function, y = log 10 x,
is used in many applications. Base 10 logarithms are called common logarithms. Common logarithms are usually written without the subscript 10. log 10 x = log x, x > 0 Most scientific calculators have a
Tennessee Curriculum Standards ✔ 3103.3.13 Solve problems using exponential functions requiring the use of logarithms for their solutions. ✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
key for evaluating common logarithms.
Example 1 Find Common Logarithms Use a calculator to evaluate each expression to the nearest tenthousandth. a. log 5 5
KEYSTROKES:
.6989700043
log 5 ≈ 0.6990 b. log 0.3 0.3 log 0.3 ≈ 0.5229 KEYSTROKES:
.5228787453
GuidedPractice 1A. log 7
1B. log 0.5
The common logarithms of numbers that differ by integral powers of ten are closely related. Remember that a logarithm is an exponent. For example, in the equation y = log x, y is the power to which 10 is raised to obtain the value of x. log x = y log 1 = 0 log 10 = 1 log 10 m = m
516  Lesson 86
→
means since since since
→
10 y = x 10 0 = 1 10 1 = 10 10 m = 10 m
Common logarithms are used in the measurement of sound. Soft recorded music is about 36 decibels (dB).
RealWorld Example 2 Solve Logarithmic Equations I ROCK CONCERT The loudness L, in decibels, of a sound is L = 10 log _ m , where I is the intensity of the sound and m is the minimum intensity of sound detectable by the human ear. Residents living several miles from a concert venue can hear the music at an intensity of 66.6 decibels. How many times the minimum intensity of sound detectable by the human ear was this sound, if m is defined to be 1? I L = 10 log _ m
66.6 = 10 log _I
Replace L with 66.6 and m with 1.
6.66 = log I
Divide each side by 10 and simplify.
1
RealWorldLink Acoustical Engineer Acoustical engineers are concerned with reducing unwanted sounds, noise control, and making useful sounds. Examples of useful sounds are ultrasound, sonar, and sound reproduction. Employment in this field requires a minimum of a bachelor’s degree.
Original equation
I = 10 6.66
Exponential form
I ≈ 4,570,882
Use a calculator.
The sound heard by the residents was approximately 4,570,000 times the minimum intensity of sound detectable by the human ear.
GuidedPractice 2. EARTHQUAKES The amount of energy E in ergs that an earthquake releases is related to its Richter scale magnitude M by the equation log E = 11.8 + 1.5M. Use the equation to find the amount of energy released by the 2004 Sumatran earthquake, which measured 9.0 on the Richter scale and led to a tsunami.
If both sides of an exponential equation cannot easily be written as powers of the same base, you can solve by taking the logarithm of each side.
Example 3 Solve Exponential Equations Using Logarithms Solve 4 x = 19. Round to the nearest tenthousandth. 4 x = 19 x
log 4 = log 19 x log 4 = log 19 log 19 log 4
Original equation Property of Equality for Logarithmic Functions Power Property of Logarithms
x=_
Divide each side by log 4.
x ≈ 2.1240
Use a calculator.
The solution is approximately 2.1240. CHECK You can check this answer graphically by using a graphing calculator. Graph the line y = 4 x and the line y = 19. Then use the CALC menu to find the intersection of the two graphs. The intersection is very close to the answer that was obtained algebraically. ✓ [10, 10] scl: 1 by [5, 25] scl: 1
GuidedPractice 3A. 3 x = 15
3B. 6 x = 42
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The same strategies that are used to solve exponential equations can be used to solve exponential inequalities.
StudyTip
Example 4 Solve Exponential Inequalities Using Logarithms
Solving Inequalities Remember that the direction of an inequality must be switched if each side is multiplied or divided by a negative number. Since 5 log 3  log 7 > 0, the inequality does not change.
Solve 3 5y < 7 y  2. Round to the nearest tenthousandth. 3 5y < 7 y  2
Original inequality
log 3 5y < log 7 y  2
Property of Inequality for Logarithmic Functions
5y log 3 < (y  2) log 7
Power Property of Logarithms
5y log 3 < y log 7  2 log 7
Distributive Property
5y log 3  y log 7 < 2 log 7
Subtract y log 7 from each side.
y(5 log 3  log 7) < 2 log 7
Distributive Property
2 log 7 y < __ 5 log 3  log 7
{y  y < 1.0972}
Divide each side by 5 log 3  log 7. Use a calculator.
CHECK Test y = 2. 3 5y < 7 y  2 3
5(2)
7
Original inequality
(2)  2
Replace y with 2.
3 10 7 4
Simplify.
1 1 _ <_ ✓ 59,049 2401
Negative Exponent Property
GuidedPractice Solve each inequality. Round to the nearest tenthousandth. 4A. 3 2x ≥ 6 x + 1
2
4B. 4 y < 5 2y + 1
Change of Base Formula The Change of Base Formula allows you to write equivalent logarithmic expressions that have different bases.
KeyConcept Change of Base Formula For all positive numbers a, b, and n, where a ≠ 1 and b ≠ 1,
Symbols
log n log b a
b log a n = _ .
log 11 log 10 3
10 log 3 11 = _
Example
Math HistoryLink John Napier (1550–1617) John Napier was a Scottish mathematician and theologian who began the use of logarithms to aid in calculations. He is also known for popularizing the use of the decimal point.
← log base b of original number ← log base b of old base
To prove this formula, let log a n = x. ax = n x
log b a = log b n x log b a = log b n log n
b x=_
log b a
log n
b log a n = _
log b a
518  Lesson 86  Common Logarithms
Definition of logarithm Property of Equality for Logarithmic Functions Power Property of Logarithms Divide each side by log b a. Replace x with log a n.
The Change of Base Formula makes it possible to evaluate a logarithmic expression of any base by translating the expression into one that involves common logarithms.
Example 5 Change of Base Formula Express log 3 20 in terms of common logarithms. Then round to the nearest tenthousandth. log
20
10 log 3 20 = _
log 10 3
≈ 2.7268
Change of Base Formula Use a calculator.
GuidedPractice 5. Express log 6 8 in terms of common logarithms. Then round to the nearest tenthousandth.
Check Your Understanding Example 1
Use a calculator to evaluate each expression to the nearest tenthousandth. 1. log 5
Example 2
Example 3
2. log 21
4. log 0.7
Solve each equation. Round to the nearest tenthousandth. 7. 2.1 a + 2 = 8.25
2
9 11 b  3 = 5 b
8. 7 x = 20.42
Solve each inequality. Round to the nearest tenthousandth. 11. 6 p  1 ≤ 4 p
10. 5 4n > 33 Example 5
3. log 0.4
5. SCIENCE The amount of energy E in ergs that an earthquake releases is related to its Richter scale magnitude M by the equation log E = 11.8 + 1.5M. Use the equation to find the amount of energy released by the 1960 Chilean earthquake, which measured 8.5 on the Richter scale.
6. 6 x = 40 Example 4
= StepbyStep Solutions begin on page R20.
Express each logarithm in terms of common logarithms. Then approximate its value to the nearest tenthousandth. 12. log 3 7
13. log 4 23
14. log 9 13
15. log 2 5
Practice and Problem Solving Example 1
Example 2
Extra Practice begins on page 947.
Use a calculator to evaluate each expression to the nearest tenthousandth. 16. log 3
17. log 11
18. log 3.2
19. log 8.2
20. log 0.9
21. log 0.04
22. AUTO REPAIR Loretta had a new muffler installed on her car. The noise level of the engine dropped from 85 decibels to 73 decibels. a. How many times the minimum intensity of sound detectable by the human ear was the car with the old muffler, if m is defined to be 1? b. How many times the minimum intensity of sound detectable by the human ear is the car with the new muffler? Find the percent of decrease of the intensity of the sound with the new muffler. connectED.mcgrawhill.com
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Example 3
Solve each equation. Round to the nearest tenthousandth. 23. 8 x = 40
24. 5 x = 55
25. 2.9 a  4 = 8.1
26. 9 b  1 = 7 b
2
27. 13 x = 33.3 Example 4
Example 5
B
2
28. 15 x = 110
Solve each inequality. Round to the nearest tenthousandth. 29. 6 3n > 36
30. 2 4x ≤ 20
31. 3 y  1 ≤ 4 y
32. 5 p  2 ≥ 2 p
Express each logarithm in terms of common logarithms. Then approximate its value to the nearest tenthousandth. 33. log 7 18
34. log 5 31
35. log 2 16
36. log 4 9
37. log 3 11
38. log 6 33
39 PETS The number n of pet owners in thousands after t years can be modeled by n = 35[log 4 (t + 2)]. Let t = 0 represent 2000. Use the Change of Base Formula to solve the following questions. a. How many pet owners were there in 2010? b. How long until there are 80,000 pet owners? When will this occur? 40. GRIZZLY BEARS Five years ago the grizzly bear population in a certain national park was 325. Today it is 450. Studies show that the park can support a population of 750. a. What is the average annual rate of growth in the population if the grizzly bears reproduce once a year? b. How many years will it take to reach the maximum population if the population growth continues at the same average rate? Solve each equation or inequality. Round to the nearest tenthousandth. 41. 3 x = 40
42. 5 3p = 15
43. 4 n + 2 = 14.5
44. 8 z  4 = 6.3
45. 7.4 n  3 = 32.5
46. 3.1 y  5 = 9.2
47. 5 x ≥ 42
48. 9 2a < 120
49. 3 4x ≤ 72
50. 7 2n > 52 4n + 3
51. 6 p ≤ 13 5  p
52. 2 y + 3 ≥ 8 3y
Express each logarithm in terms of common logarithms. Then approximate its value to the nearest tenthousandth. 53. log 4 12
54. log 3 21
55. log 8 2 57. log 5 (2.7)
56. log 6 7 2
58. log 7 √ 5
59. MUSIC A musical cent is a unit in a logarithmic scale of relative pitch or intervals. One octave is equal to 1200 cents. The formula n = 1200 log 2 _a can be used to b determine the difference in cents between two notes with frequencies a and b. a. Find the interval in cents when the frequency changes from 443 Hertz (Hz) to 415 Hz. b. If the interval is 55 cents and the beginning frequency is 225 Hz, find the final frequency.
(
520  Lesson 86  Common Logarithms
)
Solve each equation. Round to the nearest tenthousandth. 2
C
2
3
60. 10 x = 60
61 4 x
63. 8 2x – 4 = 4 x + 1
64. 16 x = √ 4x + 3
= 16
62. 9 6y – 2 = 3 3y + 1 65. 2 y = √ 3y  1
66. ENVIRONMENTAL SCIENCE An environmental engineer is testing drinking water wells in coastal communities for pollution, specifically unsafe levels of arsenic. The safe standard for arsenic is 0.025 parts per million (ppm). Also, the pH of the arsenic level should be less than 9.5. The formula for hydrogen ion concentration is pH = log H. (Hint: 1 kilogram of water occupies approximately 1 liter. 1 ppm = 1 mg/kg.) a. Suppose the hydrogen ion concentration of a well is 1.25 × 10 11. Should the environmental engineer be worried about too high an arsenic content? b. The environmental engineer finds 1 milligram of arsenic in a 3 liter sample, is the well safe? c. What is the hydrogen ion concentration that meets the troublesome pH level of 9.5? 67.
MULTIPLE REPRESENTATIONS In this problem, you will solve the exponential equation 4 x = 13. a. Tabular Enter the function y = 4x into a graphing calculator, create a table of values for the function, and scroll through the table to find x when y = 13. b. Graphical Graph y = 4x and y = 13 on the same screen. Use the intersect feature to find the point of intersection. c. Numerical Solve the equation algebraically. Do all of the methods produce the same result? Explain why or why not.
H.O.T. Problems
Use HigherOrder Thinking Skills
68. ERROR ANALYSIS Sam and Rosamaria are solving 4 3p = 10. Is either of them correct? Explain your reasoning.
Sam
Rosamaria
4 = 10
4 3p = 10
log 4 3p = log 10
log 4 3p = log 10
p log 4 = log 10
3p log 4 = log 10
log 10 p=_
p=_
3p
log 4
log 10 3 log 4
69. CHALLENGE Solve log √a 3 = log a x for x and explain each step. log 9
5 70. REASONING Write _ as a single logarithm.
log 5 3
71. PROOF Find the values of log 3 27 and log 27 3. Make and prove a conjecture about the relationship between log a b and log b a. 72. WRITING IN MATH Explain how exponents and logarithms are related. Include examples like how to solve a logarithmic equation using exponents and how to solve an exponential equation using logarithms. connectED.mcgrawhill.com
521
SPI 3103.3.6, SPI 3103.3.8, SPI 3108.4.12, SPI 3103.3.10
Standardized Test Practice 73. Which expression represents f[g(x)] if f(x) = x 2 + 4x + 3 and g(x) = x  5? A x 2 + 4x  2
75. GEOMETRY If the surface area of a cube is increased by a factor of 9, what is the change in the length of the sides of the cube?
B x 2  6x + 8
F The length is 2 times the original length.
C x 2  9x + 23
G The length is 3 times the original length.
D x 2  14x + 6
H The length is 6 times the original length. J The length is 9 times the original length.
74. EXTENDED RESPONSE Colleen rented 3 documentaries, 2 video games, and 2 movies. The charge was $16.29. The next week, she rented 1 documentary, 3 video games, and 4 movies for a total charge of $19.84. The third week she rented 2 documentaries, 1 video game, and 1 movie for a total charge of $9.14. a. Write a system of equations to determine the cost to rent each item. b. What is the cost to rent each item?
76. SAT/ACT Which of the following most accurately describes the translation of the graph y = (x + 4) 2  3 to the graph of y = (x  1) 2 + 3? A down 1 and to the right 3 B down 6 and to the left 5 C up 1 and to the left 3 D up 1 and to the right 3 E up 6 and to the right 5
Spiral Review Solve each equation. Check your solutions. (Lesson 85) 1 77. log 5 7 + _ log 5 4 = log 5 x
78. 2 log 2 x  log 2 (x + 3) = 2
16 79. log 6 48  log 6 _ + log 6 5 = log 6 5x
80. log 10 a + log 10 (a + 21) = 2
2
5
Solve each equation or inequality. (Lesson 84) 1 81. log 4 x = _
82. log 81 729 = x
83. log 8 (x 2 + x) = log 8 12
84. log 8 (3y  1) < log 8 (y + 5)
2
85. SAILING The area of a triangular sail is 16x 4  60x 3  28x 2 + 56x  32 square meters. The base of the triangle is x  4 meters. What is the height of the sail? (Lesson 62) 86. HOME REPAIR Mr. Turner is getting new locks installed. The locksmith charges $85 for the service call, $25 for each door, and $30 for each lock. (Lesson 24) a. Write an equation that represents the cost for x number of doors. b. Mr. Turner wants the front, side, back, and garage door locks changed. How much will this cost?
Skills Review Write an equivalent exponential equation. (Lesson 83) 87. log 2 5 = x
88. log 4 x = 3
89. log 5 25 = 2
90. log 7 10 = x
91. log 6 x = 4
92. log 4 64 = 3
522  Lesson 86  Common Logarithms
Graphing Technology Lab
Solving Logarithmic Equations and Inequalities You have solved logarithmic equations algebraically. You can also solve logarithmic equations by graphing or by using a table. The TI83/84 Plus has y = log10 x as a builtin function. Enter to view this graph. To graph logarithmic functions with bases other than 10, you must use the Change of Base Formula, loga n =
log n _ . b
logb a
Tennessee Curriculum Standards ✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
y = log10x
[2, 8] scl: 1 by [10, 10] scl: 1
Activity 1 Solve log2 (6x  8) = log3 (20x + 1). Step 1 Graph each side of the equation. Graph each side of the equation as a separate function. Enter log2 (6x  8) as Y1 and log3 (20x + 1) as Y2. Then graph the two equations. 6
KEYSTROKES:
20
8 1
2 3 [2, 8] scl: 1 by [2, 8] scl: 1
Step 2 Use the intersect feature. Use the intersect feature on the CALC menu to approximate the ordered pair of the point at which the curves intersect. The calculator screen shows that the xcoordinate of the point at which the curves intersect is 4. Therefore, the solution of the equation is 4.
Step 3 Use the TABLE feature.
[2, 8] scl: 1 by [2, 8] scl: 1
Examine the table to find the xvalue for which the yvalues for the graphs are equal. At x = 4, both functions have a yvalue of 4. Thus, the solution of the equation is 4.
You can use a similar procedure to solve logarithmic inequalities using a graphing calculator. (continued on the next page) connectED.mcgrawhill.com
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Graphing Technology Lab
Solving Logarithmic Equations and Inequalities Continued Activity 2 Solve log4 (10x + 1) < log5 (16 + 6x). Step 1 Enter the inequalities. Rewrite the problem as a system of inequalities. The first inequality is log4 (10x + 1) < y or y > log4 (10x + 1). Since this inequality includes the greater than symbol, shade above the curve. First enter the boundary and then use the arrow and choose the shade above icon,
keys to
.
The second inequality is y < log5 (16 + 6x). Shade below the curve since this inequality contains less than. 10
KEYSTROKES:
1
4
16
6
5
Step 2 Graph the system. KEYSTROKES:
The left boundary of the solution set is where the first inequality is undefined. It is undefined for 10x + 1 ≤ 0. 10x + 1 ≤ 0 10x ≤ 1
[2, 4] scl: 1 by [2, 4] scl: 1
1 x ≤ _ 10
Use the calculator’s intersect feature to find the right boundary. You can conclude that the solution set is {x  0.1 < x < 1.5}. Step 3 Use the TABLE feature to check your solution. Start the table at 0.1 and show xvalues in increments of 0.1. Scroll through the table. KEYSTROKES:
[TBLSET] 0.1
0.1
[TABLE]
The table confirms the solution of the inequality is {x  0.1 < x < 1.5}.
Exercises Solve each equation or inequality. Check your solution. 1. log2 (3x + 2) = log3 (12x + 3)
2. log6 (7x + 1) = log4 (4x  4)
3. log2 3x = log3 (2x + 2)
4. log10 (1  x) = log5 (2x + 5)
5. log4 (9x + 1) > log3 (18x  1)
6. log3 (3x  5) ≥ log3 (x + 7)
7. log5 (2x + 1) < log4 (3x  2)
8. log2 2x ≤ log4 (x + 3)
524  Extend 86  Graphing Technology Lab: Solving Logarithmic Equations and Inequalities
Base e and Natural Logarithms Then
Now
Why?
You worked with common logarithms.
1
Evaluate expressions involving the natural base and natural logarithm.
2
Solve exponential equations and inequalities using natural logarithms.
The St. Louis Gateway Arch in Missouri is in the form of an inverted catenary curve. A catenary curve directs the force of its weight along itself, so that: • if a rope or chain is hanging, it is pulled into that shape, and, • if a catenary is standing upright, it can support itself.
(Lesson 86)
NewVocabulary natural base, e natural base exponential function natural logarithm
1
The equation for the catenary curve involves e, a special number that appears throughout mathematics and science.
Base e and Natural Logarithms Like π and √2 , the number e is an irrational
number. The value of e is 2.71828… . It is referred to as the natural base, e. An exponential function with base e is called a natural base exponential function.
KeyConcept Natural Base Functions The function f (x ) = e x is used to model continuous exponential growth. The function f (x ) = e x is used to model continuous exponential decay.
Tennessee Curriculum Standards
The inverse of a natural base exponential function is called the natural logarithm. This logarithm can be written as log e x, but is more often abbreviated as ln x.
✔ 3103.3.13 Solve problems using exponential functions requiring the use of logarithms for their solutions. ✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
f(x) f (x) = e x
f(x) f (x) = e x
(1, e )
(1, e )
(0, 1) 0
(0, 1)
(e, 1) (1, 0)
x
0( 1, 0) (e, 1)
f (x) = ln x
&YQPOFOUJBM(SPXUI
x
f (x) = ln x
&YQPOFOUJBM%FDBZ
You can write an equivalent base e exponential equation for a natural logarithmic equation by using the fact that ln x = log e x. ln 4 = x → log e 4 = x → e x = 4
Example 1 Write Equivalent Expressions Write each exponential equation in logarithmic form. a. e x = 8 ex = 8 →
log e 8 = x ln 8 = x
b. e 5 = x e 5 = x → log e x = 5 ln x = 5
GuidedPractice 1A. e x = 9
1B. e 7 = x
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You can also write an equivalent natural logarithm equation for a natural base e exponential equation. e x = 12
→
log e 12 = x
→
ln 12 = x
Example 2 Write Equivalent Expressions Write each logarithmic equation in exponential form. a. ln x ≈ 0.7741
b. ln 10 = x
ln x ≈ 0.7741 → log e x = 0.7741 x≈e
ln 10 = x
→ log e 10 = x 10 = e x
0.7741
GuidedPractice 2A. ln x ≈ 2.1438
2B. ln 18 = x
The properties of logarithms you learned in Lesson 85 also apply to the natural logarithms. The logarithmic expressions below can be simplified into a single logarithmic term.
Example 3 Simplify Expressions with e and the Natural Log
StudyTip
Write each expression as a single logarithm.
Simplifying When you simplify logarithmic expressions, verify that the logarithm contains no operations and no powers.
a. 3 ln 10  ln 8 3 ln 10  ln 8 = ln 10 3  ln 8
Power Property of Logarithms
3
10 = ln _
Quotient Property of Logarithms
= ln 125
Simplify.
= ln 5 3
5 3 = 125
= 3 ln 5
Power Property of Logarithms
8
CHECK Use a calculator to verify the solution. KEYSTROKES: 3 10 8 KEYSTROKES:
3
4.828313737
4.828313737
5
_
b. ln 40 + 2 ln 1 + ln x 2
1 1 ln 40 + 2 ln _ + ln x = ln 40 + ln _ + ln x 2
4
Power Property of Logarithms
1 = ln 40 · _ ·x
Product Property of Logarithms
= ln 10x
Simplify.
(
4
)
GuidedPractice 3A. 6 ln 8  2 ln 4
3B. 2 ln 5 + 4 ln 2 + ln 5y
StudyTip Look Back Refer to Lesson 72 to review inverse functions.
Because the natural base and natural log are inverse functions, they can be used to undo or eliminate each other. e ln x = x
526  Lesson 87  Base e and Natural Logarithms
ln e x = x
2
Equations and Inequalities with e and ln Equations and inequalities involving
base e are easier to solve by using natural logarithms rather than by using common logarithms, because ln e = 1.
Example 4 Solve Base e Equations Solve 4e 2x  5 = 3. Round to the nearest tenthousandth. 4e 2x  5 = 3 4e 2x = 8 e 2x = 2 ln e 2x = ln 2 2x = ln 2
Calculators Most calculators have an e x and LN key for evaluating natural base and natural log expressions.
Add 5 to each side. Divide each side by 4. Property of Equality for Logarithms ln e x = x
ln 2 x=_
Divide each side by 2.
x ≈ 0.3466
Use a calculator.
2
StudyTip
Original equation
2
KEYSTROKES:
2
.34657359
GuidedPractice Solve each equation. Round to the nearest tenthousandth. 4B. 4e x + 8 = 17
4A. 3e 4x  12 = 15
Just like the natural logarithm can be used to eliminate e x, the natural base exponential function can eliminate ln x.
Example 5 Solve Natural Log Equations and Inequalities Solve each equation or inequality. Round to the nearest tenthousandth. a. 3 ln 4x = 24 3 ln 4x = 24 ln 4x = 8 e ln 4x = e 8 4x = e 8 8
Original equation Divide each side by 3. Property of Equality for Exponential Functions e ln x = x
e x=_
Divide each side by 4.
x ≈ 745.2395
Use a calculator.
4
b. ln (x  8) 4 < 4 ln (x  8) 4 < 4 4
e ln (x  8) < e 4 (x  8) 4 < e 4 x8
Original equation Write each side using exponents and base e. e ln x = x Property of Equality for Exponential Functions Add 8 to each side. Use a calculator.
GuidedPractice Solve each equation or inequality. Round to four decimal places. 5A. 5 ln 6x = 8
5B. ln (2x  3) 3 > 6
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Interest that is compounded continuously can be found using e.
KeyConcept Continuosly Compounded Interest Calculate continuously compounded interest using the following formula. A = Pe rt, where A is the amount in the account after t years, P is the principal amount invested, and r is the annual interest rate.
RealWorld Example 6 Solve Base e Inequalities FINANCIAL LITERACY When Angelina was born, her grandparents deposited $3000 into a college savings account paying 4% interest compounded continuously.
RealWorldLink The average cost of tuition at fouryear public colleges is about $6185 per year. Source: CNN
a. Assuming there are no deposits or withdrawals from the account, what will the balance be after 10 years? A = Pe rt Continuous Compounding Formula (0.04)(10) = 3000e P = 3000, r = 0.04, and t = 10 0.4 = 3000e Simplify. ≈ 4475.47 Use a calculator. The balance will be $4475.47. b. How long will it take the balance to reach at least $10,000? A < Pe rt Continuous Compounding Formula 10,000 < 3000e (0.04)t P = 3000, r = 0.04, and A = 10,000 10 _ < e 0.04t
3 10 _ ln < ln e 0.04t 3 10 ln _ < 0.04t 3 10 ln _ 3 _
0.04
Divide each side by 3000. Property of Equality of Logarithms ln e x = x Divide each side by 0.04.
30.099 < t Use a calculator. It will take about 30 years to reach at least $10,000. c. If her grandparents want Angelina to have $10,000 after 18 years, how much would they need to invest? 10,000 = Pe (0.04)18 A = 10,000, r = 0.04, and t = 18
StudyTip Rounding Not rounding until the very end will provide a more accurate answer.
10,000 _ =P e 0.72
4867.52 ≈ P
Divide each side by e 0.72. Use a calculator.
They need to invest $4867.52.
GuidedPractice 6. Use the information in Example 6 to answer the following. A. If they invested $8000 at 3.75% interest compounded continuously, how much money would be in the account in 30 years? B. If they could only deposit $10,000 in the account above, at what rate would the account need to grow in order for Angelina to have $30,000 in 18 years? C. If Angelina’s grandparents found an account that paid 5% compounded continuously and wanted her to have $30,000 after 18 years, how much would they need to deposit?
528  Lesson 87  Base e and Natural Logarithms
Check Your Understanding
= StepbyStep Solutions begin on page R20.
Examples 1–2 Write an equivalent exponential or logarithmic function.
Example 3
Example 4
Example 5
Example 6
1. e x = 30
2. ln x = 42
3. e 3 = x
4. ln 18 = x
Write each as a single logarithm. 5. 3 ln 2 + 2 ln 4
6. 5 ln 3  2 ln 9
7. 3 ln 6 + 2 ln 9
8. 3 ln 5 + 4 ln x
Solve each equation. Round to the nearest tenthousandth. 9. 5e x  24 = 16
10. 3e x + 9 = 4
11. 3e 3x + 4 = 6
12. 2e x  3 = 8
Solve each equation or inequality. Round to the nearest tenthousandth. 13. ln 3x = 8
14. 4 ln 2x = 26
15. ln (x + 5) 2 < 6
16. ln (x  2) 3 > 15
17. e x > 29
18. 5 + e x > 14
19. SCIENCE A virus is spreading through a computer network according to the formula v(t) = 30e 0.1t, where v is the number of computers infected and t is the time in minutes. How long will it take the virus to infect 10,000 computers?
Practice and Problem Solving
Extra Practice begins on page 947.
Examples 1–2 Write an equivalent exponential or logarithmic function.
Example 3
20. e x = 8
21. e 5x = 0.1
22. ln 0.25 = x
23. ln 5.4 = x
24. e x  3 = 2
25. ln (x + 4) = 36
26. e 2 = x 6
27. ln e x = 7
Write each as a single logarithm. 28. ln 125  2 ln 5 1 + 5 ln 2 31 7 ln _ 2
Example 4
29. 3 ln 10 + 2 ln 100 32. 8 ln x  4 ln 5
1 1 30. 4 ln _  6 ln _ 3
9
2
33. 3 ln x + 4 ln 3
Solve each equation. Round to the nearest tenthousandth. 34. 6e x  3 = 35
35. 4e x + 2 = 180
36. 3e 2x  5 = 4
37. 2e 3x + 19 = 3
38. 6e 4x + 7 = 4
39. 4e x + 9 = 2
Examples 5–6 40. FINANCIAL LITERACY The value of a certain car depreciates according to v(t) = 18500e 0.186t, where t is the number of years after the car is purchased new. a. What will the car be worth in 18 months? b. When will the car be worth half of its original value? c. When will the car be worth less than $1000? Solve each inequality. Round to the nearest tenthousandth. 41. e x ≤ 8.7
42. e x ≥ 42.1
43. ln (3x + 4) 3 > 10
44. 4 ln x 2 < 72
45. ln (8x 4) > 24
46. 2 [ln (x  6) 1] ≤ 6 connectED.mcgrawhill.com
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B
47 FINANCIAL LITERACY Use the formula for continuously compounded interest. a. If you deposited $800 in an account paying 4.5% interest compounded continuously, how much money would be in the account in 5 years? b. How long would it take you to double your money? c. If you want to double your money in 9 years, what rate would you need? d. If you want to open an account that pays 4.75% interest compounded continuously and have $10,000 in the account 12 years after your deposit, how much would you need to deposit? Write the expression as a sum or difference of logarithms or multiples of logarithms. 48. ln 12x 2
16 49. ln _ 125
5 50. ln √x3
51. ln xy 4z 3
Use the natural logarithm to solve each equation. 52. 8 x = 24
C
53. 3 x = 0.4
54. 2 3x = 18
55. 5 2x = 38
56. SCIENCE Newton’s Law of Cooling, which can be used to determine how fast an object will cool in given surroundings, is represented by T(t) = T s + (T 0  T s)e kt, where T 0 is the initial temperature of the object, T s is the temperature of the surroundings, t is the time in minutes, and k is a constant value that depends on the type of object. a. If a cup of coffee with an initial temperature of 180º is placed in a room with a temperature of 70º and the coffee cools to 140º after 10 minutes, find the value of k. b. Use this value of k to determine the temperature of the coffee after 20 minutes. c. When will the temperature of the coffee reach 75º? 57.
MULTIPLE REPRESENTATIONS In this problem, you will use f(x) = e x and g(x) = ln x. a. Graphical Graph both functions and their axis of symmetry, y = x, for 5 ≤ x ≤ 5. Then graph a(x) = e x on the same graph. b. Analytical The graphs of a(x) and f(x) are reflections along which axis? What function would be a reflection of f(x) along the other axis? c. Graphical Determine the two functions that are reflections of g(x). Graph these new functions. d. Verbal We know that f(x) and g(x) are inverses. Are any of the other functions that we have graphed inverses as well? Explain your reasoning.
H.O.T. Problems
Use HigherOrder Thinking Skills
58. CHALLENGE Solve 4 x  2 x + 1 = 15 for x. 59. PROOF Prove ln ab = ln a + ln b for natural logarithms. 60. REASONING Determine whether x > ln x is sometimes, always, or never true. Explain your reasoning. 61. OPEN ENDED Express the value 3 using e x and the natural log. 62.
E WRITING IN MATH Explain how the natural log can be used to solve a natural base exponential function.
530  Lesson 87  Base e and Natural Logarithms
SPI 3103.3.10, SPI 3102.3.5
Standardized Test Practice 63. Given the function y = 2.34x + 11.33, which statement best describes the effect of moving the graph down two units? A B C D
65. Solve ⎪2x  5⎥ = 17. F G H J
The xintercept decreases. The yintercept decreases. The xintercept remains the same. The yintercept remains the same.
6, 11 6, 11 6, 11 6, 11
66. A local pet store sells rabbit food. The cost of two 5pound bags is $7.99. The total cost c of purchasing n bags can be found by—
64. GRIDDED RESPONSE Aidan sells wooden picture frames over the Internet. He purchases materials for $85 and pays $19.95 for his website. If he charges $15 each, how many frames will he need to sell in order to make a profit of at least $270?
A B C D
multiplying n by c. multiplying n by 5. multiplying n by the cost of 1 bag. dividing n by c.
Spiral Review Solve each equation or inequality. Round to the nearest tenthousandth. (Lesson 86) 2
67. 2 x = 53
68. 2.3 x = 66.6
69. 3 4x  7 < 4 2x + 3
70. 6 3y = 8 y  1
71. 12 x  5 ≥ 9.32
72. 2.1 x  5 = 9.32
73. SOUND Use the formula L = 10 log 10 R, where L is the loudness of a sound and R is the sound’s relative intensity. Suppose the sound of one alarm clock is 80 decibels. Find out how much louder 10 alarm clocks would be than one alarm clock. (Lesson 85)
Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. (Lesson 66) 74. x 3 + 5x 2 + 8x + 4; x + 1
75. x 3 + 4x 2 + 7x + 6; x + 2
76. CRAFTS Mrs. Hall is selling crocheted items. She sells large afghans for $60, baby blankets for $40, doilies for $25, and pot holders for $5. She takes the following number of items to the fair: 12 afghans, 25 baby blankets, 45 doilies, and 50 pot holders. (Lesson 43) a. Write an inventory matrix for the number of each item and a cost matrix for the price of each item. b. Suppose Mrs. Hall sells all of the items. Find her total income as a matrix.
Skills Review Solve each equation. (Lesson 92) 77. 2 3x + 5 = 128 1 80. _
(7)
y3
= 343
m
1 78. 5 n  3 = _
1 79. _
81. 10 x  1 = 100 2x  3
82. 36 2p = 216 p  1
25
(9)
= 81 m + 4
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Spreadsheet Lab
Compound Interest You can use a spreadsheet to organize and display data. A spreadsheet is an easy way to track the amount of interest earned over a period of time. Compound interest is earned not only on the original amount, but also on any interest that has been added to the principal.
Tennessee Curriculum Standards ✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
Activity Find the total amount of money after 5 years if you deposit $100 at 7% compounded annually. Step 1 Label your columns as shown. The period is one year. Enter the starting values and the rate. Step 2 Each row will be generated using formulas. Enter the formulas as shown.
4BWJOHT"DDPVOU "
#
$
%
&
&OEPG1FSJPE 1SJODJQBM *OUFSFTU #BMBODF 3BUFQFS1FSJPE =#& =% ="+ =$+% 4IFFU
4IFFU
4IFFU
Step 3 Use the FILL DOWN function to fill 4 additional rows.
4BWJOHT"DDPVOU "
#
&OEPG1FSJPE 4IFFU
$
1SJODJQBM *OUFSFTU 4IFFU
%
&
#BMBODF 3BUFQFS1FSJPE
4IFFU
If you deposit $100 at 7% annual interest for 5 years, you will have $140.26 at the end of the 5 years.
Exercises Find the total balance for each situation. 1. deposit $500 for 7 years at 5%
2. deposit $1000 for 5 years at 6%
3. deposit $200 for 2 years at 10%
4. deposit $800 for 3 years at 8%
5. borrow $10,000 for 5 years at 5.05%
6. borrow $25,000 for 30 years at 8%
532  Explore 88  Spreadsheet Lab: Compound Interest
Using Exponential and Logarithmic Functions Then
Now
Why?
You used exponential growth and decay formulas.
1
Use logarithms to solve problems involving exponential growth and decay.
2
Use logarithms to solve problems involving logistic growth.
The ancient footprints of Acahualinca, discovered in Managua, Nicaragua, are believed to be the oldest human footprints in the world. Using carbon dating, scientists estimate that these footprints are 6000 years old.
(Lesson 81)
NewVocabulary rate of continuous growth rate of continuous decay logistic growth model
1
Exponential Growth and Decay Scientists and researchers frequently use alternate forms of the growth and decay formulas that you learned in Lesson 81.
KeyConcept Exponential Growth and Decay Exponential Growth
Tennessee Curriculum Standards ✔ 3103.3.13 Solve problems using exponential functions requiring the use of logarithms for their solutions. ✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
Exponential Decay
Exponential growth can be modeled by the function f (x ) = ae kt,
Exponential decay can be modeled by the function f (x ) = ae kt,
where a is the initial value, t is time in years, and k is a constant representing the rate of continuous growth.
where a is the initial value, t is time in years, and k is a constant representing the rate of continuous decay.
RealWorld Example 1 Exponential Decay SCIENCE The halflife of a radioactive substance is the time it takes for half of the atoms of the substance to disintegrate. The halflife of Carbon14 is 5730 years. Determine the value of k and the equation of decay for Carbon14. If a is the initial amount of the substance, then the amount y that remains after 5730 years can be represented by 12a or 0.5a. y = ae kt 0.5a = ae
k(5730)
0.5 = e 5730k ln 0.5 = ln e
5730k
ln 0.5 = 5730k ln 0.5 _ =k 5730
0.00012 ≈ k
Exponential Decay Formula y = 0.5a and t = 5730 Divide each side by a. Property of Equality for Logarithmic Functions ln e x = x Divide each side by 5730. Use a calculator.
Thus, the equation for the decay of Carbon14 is y = ae 0.00012t.
GuidedPractice 1. The halflife of Plutonium239 is 24,000 years. Determine the value of k.
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Now that the value of k for Carbon14 is known, it can be used to date fossils.
RealWorld Example 2 Carbon Dating SCIENCE A paleontologist examining the bones of a prehistoric animal estimates that they contain 2% as much Carbon14 as they would have contained when the animal was alive. a. How long ago did the animal live?
RealWorldLink The oldest modern human fossil, found in Ethiopia, is approximately 160,000 years old. Source: National Public Radio
Understand The formula for the decay of Carbon14 is y = ae 0.00012t. You want to find out how long ago the animal lived. Plan Let a be the initial amount of Carbon14 in the animal’s body. The amount y that remains after t years is 2% of a or 0.02a. Solve
y = ae 0.00012t 0.02a = ae 0.00012t 0.02 = e 0.00012t ln 0.02 = ln e 0.00012t ln 0.02 = 0.00012t ln 0.02 _ =t 0.00012
Formula for the decay of Carbon14 y = 0.02a Divide each side by a. Property of Equality for Logarithmic Functions ln e x = x Divide each side by 0.00012.
32,600 ≈ t Use a calculator. The animal lived about 32,600 years ago. Check Use the formula to find the amount of a sample remaining after 32,600 years. Use an original amount of 1.
StudyTip Carbon Dating When given a percent or fraction of decay, use an original amount of 1 for a.
y = ae 0.00012t Original equation 0.00012(32,600) = 1e a = 1 and t = 32,600 ≈ 0.02 or 2% Use a calculator. b. If prior research points to the animal being around 20,000 years old, how much Carbon14 should be in the animal? y = ae 0.00012t = 1e 0.00012(20,000) = e 2.4 = 0.09 or 9%
Formula for the decay of Carbon14 a = 1 and t = 20,000 Simplify. Use a calculator.
GuidedPractice 2. Use the information in Example 2 to answer the following. A. A specimen that originally contained 42 milligrams of Carbon14 now contains 8 milligrams. How old is the fossil? B. A wooly mammoth specimen was thought to be about 12,000 years old. How much Carbon14 should be in the animal?
The exponential growth equation y = ae kt is identical to the continuously compounded interest formula you learned in Lesson 87. Continuous Compounding
Population Growth
A = Pe rt
y = ae kt
P = initial amount A = amount at time t r = interest rate
a = initial population y = population at time t k = rate of continuous growth
534  Lesson 88  Using Exponential and Logarithmic Functions
RealWorld Example 3 Continuous Exponential Growth POPULATION In 2007, the population of Georgia was 9.36 million people. In 2000, it was 8.18 million.
ProblemSolvingTip Use a Formula When dealing with population, it is almost always necessary to use an exponential growth or decay formula.
a. Determine the value of k, Georgia’s relative rate of growth. y = ae kt Formula for continuous exponential growth 9.36 = 8.18e k(7)
y = 9.36, a = 8.18, and t = 2007  2000 or 7
9.36 _ = e 7k 8.18 9.36 ln _ = ln e 7k 8.18 9.36 ln _ = 7k 8.18
Divide each side by 8.18. Property of Equality for Logarithmic Functions ln e x = x
9.36 ln _
8.18 _ =k
Divide each side by 7.
7
0.01925 = k
Use a calculator.
Georgia’s relative rate of growth is about 0.01925 or about 2%. b. When will Georgia’s population reach 12 million people? y = ae kt Formula for continuous exponential growth 12 = 8.18e 0.01925t 1.4670 = e 0.01925t ln 1.4670 = ln e
y = 10, a = 8.18, and k = 0.01925 Divide each side by 8.18.
0.01925t
Property of Equality for Logarithmic Functions
StudyTip
ln 1.4670 = 0.01925t
ln e x = x
Rounding Error In order to avoid any errors due to rounding, do not round until the very end of your calculations.
ln 1.4670 _ =t
Divide each side by 0.01925.
0.01925
19.907 ≈ t
Use a calculator.
Georgia’s population will reach 12 million people by 2020. c. Michigan’s population in 2000 was 9.9 million and can be modeled by y = 9.9e 0.0028t. Determine when Georgia’s population will surpass Michigan’s. 8.18e 0.01925t > 9.9e 0.0028t Formula for exponential growth ln 8.18e 0.01925t > ln 9.9e 0.0028t ln 8.18 + ln e
0.01925t
> ln 9.9 + ln e
Property of Inequality for Logarithms 0.0028t
ln 8.18 + 0.01925t > ln 9.9 + 0.0028t 0.01645t > ln 9.9  ln 8.18 ln 9.9  ln 8.18 t > __ 0.01645
Product Property of Logarithms ln e x = x Subtract (0.0028t + ln 8.18) from each side. Divide each side by 0.01645.
t > 11.6 Use a calculator. Georgia’s population will surpass Michigan’s during 2012.
GuidedPractice 3. BIOLOGY A type of bacteria is growing exponentially according to the model y = 1000e kt, where t is the time in minutes. A. If there are 1000 cells initially and 1650 cells after 40 minutes, find the value of k for the bacteria. B. Suppose a second type of bacteria is growing exponentially according to the model y = 50e 0.0432t. Determine how long it will be before the number of cells of this bacteria exceed the number of cells in the other bacteria.
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Georgia’s population in Example 3. According to the graph at the right, Georgia’s population will be about one billion by the year 2130. Does this seem logical? Populations cannot grow infinitely large. There are limitations, such as food supplies, war, living space, diseases, available resources, and so on.
Population (millions)
2
y
Logistic Growth Refer to the equation representing 80 60 40 20 0
Exponential growth is unrestricted, meaning it will increase without bound. A logistic growth model, however, represents growth that has a limiting factor. Logistic models are the most accurate models for representing population growth.
40
80
120
x
160
Years since 2000
KeyConcept Logistic Growth Function Let a, b, and c be positive constants where b < 1. The logistic growth function is represented c by f (t ) = _ , where t represents time. bt 1 + ae
Example 4 Logistic Growth The population of Phoenix, Arizona, in millions can be modeled by the logistic 2.0666 function f (t ) = , where t is the number of years after 1980. 0.048t
__
a. Graph the function for 0 ≤ t ≤ 500.
Phoenix is the fifth largest city in the country and has a population of 1.5 million.
b. What is the horizontal asymptote? The horizontal asymptote is at y = 2.0666. c. Will the population of Phoenix increase indefinitely? If not, what will be their maximum population? No. The population will reach a maximum of a little less than 2.0666 million people.
StudyTip Intersections To determine where the graph intersects 1.8 on the calculator, graph y = 1.8 on the same graph and select intersection in the CALC menu.
Population (millions)
1 + 1.66e
RealWorldLink
2.7 2.4 2.1 1.8 1.5 1.2 0.9 0.6 0.3 0
f(t)
100
200
300
400
t
Number of Years After 1980
d. According to the function, when will the population of Phoenix reach 1.8 million people? The graph indicates the population will reach 1.8 million people at t ≈ 50. Replacing f (t ) with 1.8 and solving for t in the equation yields t = 50.35 years. So, the population of Phoenix will reach 1.8 million people by 2031.
GuidedPractice 4. The population of a certain species of fish in a lake after t years can be modeled by 1880 the function P (t ) = __ , where t ≥ 0. 0.037t 1 + 1.42e
A. Graph the function for 0 ≤ t ≤ 500. B. What is the horizontal asymptote? C. What is the maximum population of the fish in the lake? D. When will the population reach 1875?
536  Lesson 88  Using Exponential and Logarithmic Functions
Check Your Understanding Examples 1–2
= StepbyStep Solutions begin on page R20.
1. PALEONTOLOGY The halflife of Potassium40 is about 1.25 billion years. a. Determine the value of k and the equation of decay for Potassium40. b. A specimen currently contains 36 milligrams of Potassium40. How long will it take the specimen to decay to only 15 milligrams of Potassium40? c. How many milligrams of Potassium40 will be left after 300 million years? d. How long will it take Potassium40 to decay to one eighth of its original amount?
Example 3
2. SCIENCE A certain food is dropped on the floor and is growing bacteria exponentially according to the model y = 2e kt, where t is the time in seconds. a. If there are 2 cells initially and 8 cells after 20 seconds, find the value of k for the bacteria. b. The “5second rule” says that if a person who drops food on the floor eats it within 5 seconds, there will be no harm. How much bacteria is on the food after 5 seconds? c. Would you eat food that had been on the floor for 5 seconds? Why or why not? Do you think that the information you obtained in this exercise is reasonable? Explain.
Example 4
3. ZOOLOGY Suppose the red fox population in a restricted habitat follows the function 16,500
P(t) = __ , where t represents the time in years. 0.085t 1 + 18e
a. Graph the function for 0 ≤ t ≤ 200. b. What is the horizontal asymptote? c. What is the maximum population? d. When does the population reach 16,450?
Practice and Problem Solving Examples 1–2
Extra Practice begins on page 947.
4. SCIENCE The halflife of Rubidium87 is about 48.8 billion years. a. Determine the value of k and the equation of decay for Rubidium87. b. A specimen currently contains 50 milligrams of Rubidium87. How long will it take the specimen to decay to only 18 milligrams of Rubidium87? c. How many milligrams of Rubidium87 will be left after 800 million years? d. How long will it take Rubidium87 to decay to onesixteenth its original amount?
Example 3
5 BIOLOGY A certain bacteria is growing exponentially according to the model y = 80e kt, where t is the time in minutes. a. If there are 80 cells initially and 675 cells after 30 minutes, find the value of k for the bacteria. b. When will the bacteria reach a population of 6000 cells? c. If a second type of bacteria is growing exponentially according to the model y = 35e 0.0978t, determine how long it will be before the number of cells of this bacteria exceed the number of cells in the other bacteria.
Example 4
6. FORESTRY The population of trees in a certain forest follows the function 18000 f(t) = __ , where t is the time in years. 0.084t 1 + 16e
a. Graph the function for 0 ≤ t ≤ 100. b. When does the population reach 17500 trees? connectED.mcgrawhill.com
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B
7 PALEONTOLOGY A paleontologist finds a human bone and determines that the Carbon14 found in the bone is 85% of that found in living bone tissue. How old is the bone? 8. ANTHROPOLOGY An anthropologist has determined that a newly discovered human bone is 8000 years old. How much of the original amount of Carbon14 is in the bone? 9. RADIOACTIVE DECAY 100 milligrams of Uranium238 are stored in a container. If Uranium238 has a halflife of about 4.47 billion years, after how many years will only 10 milligrams be present? 10. POPULATION GROWTH The population of the state of Oregon has grown from 3.4 million in 2000 to 3.7 million in 2006. a. Write an exponential growth equation of the form y = ae kt for Oregon, where t is the number of years after 2000. b. Use your equation to predict the population of Oregon in 2020. c. According to the equation, when will Oregon reach 6 million people? 11. HALFLIFE A substance decays 99.9% of its total mass after 200 years. Determine the halflife of the substance.
C
12. LOGISTIC GROWTH The population in millions of the state of Ohio after 1900 can be 12.95 modeled by P(t) = _ , where t is the number of years after 1900 and k is 1 + 2.4e kt a constant. a. If Ohio had a population of 10 million in 1970, find the value of k. b. According to the equation, when will the population of Ohio reach 12 million?
13.
MULTIPLE REPRESENTATIONS In this problem, you will explore population growth. The population growth of a country follows the exponential function f(t) = 8e 0.075t or 400 the logistic function g(t) = __ . The population is measured in millions and 1 + 16 e 0.025t t is time in years. a. Graphical Graph both functions for 0 ≤ t ≤ 100. b. Analytical Determine the intersection of the graphs. What is the significance of this intersection? c. Analytical Which function is a more accurate estimate of the country’s population 100 years from now? Explain your reasoning.
H.O.T. Problems
Use HigherOrder Thinking Skills
14. OPEN ENDED Give an example of a quantity that grows or decays at a fixed rate. Write a realworld problem involving the rate and solve by using logarithms. 120,000
15. CHALLENGE Solve __ = 24e 0.055t for t. 0.015t 1 + 48e
c 16. REASONING Explain mathematically why f(t) = _ approaches, but never 1 + 60e 0.5t reaches the value of c as t → +∞.
17. OPEN ENDED Give an example of a quantity that grows logistically and has limitations to growth. Explain why the quantity grows in this manner. 18. WRITING IN MATH Summarize the differences between exponential, continuous exponential, and logistic growth.
538  Lesson 88  Using Exponential and Logarithmic Functions
SPI 3108.4.13, SPI 3103.5.8, SPI 3108.5.1, SPI 3103.1.2
Standardized Test Practice 19. Kareem is making a circle graph showing the favorite ice cream flavors of customers at his store. The table summarizes the data. What central angle should Kareem use for the section representing chocolate?
Flavor
Customers
chocolate vanilla strawberry mint chip butter pecan
35 42 7 12 4
B 14 in 2
E 254 in 2
C 32 in 2
A 35°
C 126°
B 63°
D 150°
20. PROBABILITY Lydia has 6 books on her bookshelf. Two are literature books, one is a science book, two are math books, and one is a dictionary. What is the probability that she randomly chooses a science book and the dictionary? 1 1 F _ H _ 3
12
1 G _
1 J _
4
21. SAT/ACT Peter has made a game for his daughter’s birthday party. The playing board is a circle divided evenly into 8 sectors. If the circle has a radius of 18 inches, what is the approximate area of one of the sectors? A 4 in 2 D 127 in 2
22. STATISTICS In a survey of 90 physical trainers, 15 said they went for a run at least 5 times per week. Of that group, 5 said they also swim during the week, and at least 25% of all trainers run and swim every week. Which conclusion is valid based on the information given? F The report is accurate because 15 out of 90 is 25%. G The report is accurate because 5 out of 15 is 33%, which is at least 25%. H The report is inaccurate because 5 out of 90 is only 5.6%.
15
J The report is inaccurate because no one knows if swimming is really exercising.
Spiral Review Write an equivalent exponential or logarithmic equation. (Lesson 87) 23. e 7 = y
24. e 2n  4 = 36
25. ln 5 + 4 ln x = 9
26. EARTHQUAKES The table shows the magnitudes of some major earthquakes. (Lessons 85 and 86)
a. For which two earthquakes was the intensity of one 10 times that of the other? For which two was the intensity of one 100 times that of the other? b. What would be the magnitude of an earthquake that is 1000 times as intense as the 1963 earthquake in Yugoslavia? c. Suppose you know that log 7 2 ≈ 0.3562 and log 7 3 ≈ 0.5646. Describe two different methods that you could use to approximate log 7 2.5. (You may use a calculator, of course.) Then describe how you can check your result.
Year
Location
Magnitude
1963 1970 1988 2004 2007 2010
Yugoslavia Peru Armenia Morocco Indonesia Haiti
6.0 7.8 7.0 6.4 8.4 7.0
Skills Review Solve each equation. Write in simplest form. (Lesson 13) 8 4 27. _ x=_ 15 5 9 31. _b = 18 8
6 27 28. _ n=_ 7 14 3 6 32. _y = _ 4 7
3 12 29. _ =_ a 25 5 1 33. _z = _ 6 3 10
6 30. _ = 9p 7
2 34. _ q=7 3
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Graphing Technology Lab
Cooling In this lab, you will explore the type of equation that models the change in the temperature of water as it cools under various conditions. Tennessee Curriculum Standards
Set Up the Lab • Collect a variety of containers, such as a foam cup, a ceramic coffee mug, and an insulated cup.
✔ 3103.3.16 Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems.
• Boil water or collect hot water from a tap. • Choose a container to test and ﬁll with hot water. Place the temperature probe in the cup. • Connect the temperature probe to your data collection device.
Activity
Description
Step 1 Program the device to collect 20 or more samples in 1 minute intervals. Step 2 Wait a few seconds for the probe to warm to the temperature of the water. Step 3 Press the button to begin collecting data.
Analyze the Results 1. When the data collection is complete, graph the data in a scatter plot. Use time as the independent variable and temperature as the dependent variable. Write a sentence that describes the points on the graph. 2. Use the STAT menu to find an equation to model the data you collected. Try linear, quadratic, and exponential models. Which model appears to fit the data best? Explain. 3. Would you expect the temperature of the water to drop below the temperature of the room? Explain your reasoning. 4. Use the data collection device to find the temperature of the air in the room. Graph the function y = t, where t is the temperature of the room, along with the scatter plot and the model equation. Describe the relationship among the graphs. What is the meaning of the relationship in the context of the experiment?
Make a Conjecture 5. Do you think the results of the experiment would change if you used an insulated container for the water? Repeat the experiment to verify your conjecture. 6. How might the results of the experiment change if you added ice to the water? Repeat the experiment to verify your conjecture.
540  Extend 88  Graphing Technology Lab: Cooling
Study Guide and Review Study Guide KeyConcepts
KeyVocabulary asymptote (p. 475)
logarithmic equation (p. 502)
• An exponential function is in the form y = ab , where a ≠ 0, b > 0 and b ≠ 1.
Change of Base Formula
logarithmic function (p. 493)
• Property of Equality for Exponential Functions: If b is a positive number other than 1, then b x = b y if and only if x = y.
common logarithm (p. 516)
Exponential Functions (Lessons 81 and 82) x
• Property of Inequality for Exponential Functions: If b > 1, then b x > b y if and only if x > y, and b x < b y if and only if x < y.
(p. 518)
logarithmic inequality (p. 503)
compound interest (p. 486) decay factor (p. 478)
logarithm (p. 492) logistic growth model (p. 536) natural base, e (p. 525)
exponential decay (p. 477) exponential equation (p. 485)
natural base exponential function (p. 525)
• Suppose b > 0 and b ≠ 1. For x > 0, there is a number y such that log b x = y if and only if b y = x.
exponential function (p. 475)
natural logarithm (p. 525)
exponential growth (p. 475)
rate of continuous decay
• The logarithm of a product is the sum of the logarithms of its factors.
exponential inequality (p. 487)
Logarithms and Logarithmic Functions (Lessons 83 through 86)
• The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. • The logarithm of a power is the product of the logarithm and the exponent. log n log b a
b • The Change of Base Formula: log a n = _
Natural Logarithms (Lesson 87) • Since the natural base function and the natural logarithmic function are inverses, these two can be used to “undo” each other.
Using Exponential and Logarithmic Functions (Lesson 88) • Exponential growth can be modeled by the function f (x ) = ae kt, where k is a constant representing the rate of continuous growth. • Exponential decay can be modeled by the function f (x ) = ae kt, where k is a constant representing the rate of continuous decay.
StudyOrganizer Be sure the Key Concepts are noted in your Foldable.
(p. 533)
rate of continuous growth
growth factor (p. 477)
(p. 533)
VocabularyCheck Choose a word or term from the list above that best completes each statement or phrase. 1. A function of the form f (x ) = b x where b > 1 is a(n) function. 2. In x = b y, the variable y is called the 3. Base 10 logarithms are called 4. A(n) as exponents.
of x. .
is an equation in which variables occur
5. The allows you to write equivalent logarithmic expressions that have different bases. 6. The base of the exponential function, A(t ) = a (1  r ) t, 1  r is called the . 7. The function y = log b x, where b > 0 and b ≠ 1, is called a (n ) . 8. An exponential function with base e is called the . 9. The logarithm with base e is called the 10. The number e is referred to as the
. .
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Study Guide and Review Continued LessonbyLesson Review
811 Graphing Exponential Functions
✔3103.3.11, SPI 3103.3.5, SPI 3103.3.10
(pp. 475–482)
Graph each function. State the domain and range.
Example 1
11. f (x ) = 3 x
Graph f (x ) = 2(3) x + 1. State the domain and range.
12. f (x ) = 5(2) x
13. f (x ) = 3(4) x  6 1 15. f (x ) = 3 _
(4)
x+3
14. f (x ) = 3 2x + 5 1
3 _ 2 16. f (x ) = _
5(3)
x2
+3
f(x)
0
The domain is all real numbers, and the range is all real numbers less than 1.
17. POPULATION A city with a population of 120,000 decreases at a rate of 3% annually.
x
a. Write the function that represents this situation. b. What will the population be in 10 years?
SPI 3103.3.13
822 Solving Exponential Equations and Inequalities
(pp. 485–491)
Solve each equation or inequality.
Example 2
1 18. 16 x = _
19. 3 4x = 9 3x + 7
Solve 4 3x = 32 x  1 for x.
20. 64 3n = 8 2n  3
21. 8 3  3y = 256 4y
64
1 22. 9 x  2 > _
( 81 )
x+2
4 3x = 32 x 1
(2
23. 27 3x ≤ 9 2x  1
2) 3x
= (2
6x
24. BACTERIA A bacteria population started with 5000 bacteria. After 8 hours there were 28,000 in the sample.
2 =2
Original equation
5) x  1
5x  5
Power of a Power
6x = 5x  5
Property of Equality for Exponential Functions
a. Write an exponential function that could be used to
model the number of bacteria after x hours if the number of bacteria changes at the same rate.
Rewrite so each side has the same base.
x = 5
Subtract 5x from each side.
The solution is 5.
b. How many bacteria can be expected in the sample
after 32 hours?
✔3103.1.8, ✔3103.3.17, SPI 3103.3.10
and Logarithmic Functions 833 Logarithms _
(pp. 492–499)
25. Write log 2 1 = 4 in exponential form.
Example 3
26. Write 10 2 = 100 in logarithmic form.
Evaluate log 2 64.
Evaluate each expression.
log 2 64 = y
16
27. log 4 256
64 = 2 y
1 28. log 2 _ 8
6
2 =2
Graph each function. 29. f (x ) = 2 log 10 x + 4
Let the logarithm equal y.
1 30. f (x ) = _ log _1 (x  2) 6
542  Chapter 8  Study Guide and Review
3
6=y
y
Definition of logarithm 64 = 2 6 Property of Equality for Exponential Functions
✔3103.3.16
844 Solving Logarithmic Equations and Inequalities
(pp. 502–507)
Solve each equation or inequality.
Example 4
3 31. log 4 x = _
Solve log 27 x < 2 .
2
33. log 4 x < 3
1 32. log 2 _ =x 64
34. log 5 x < 3
_ 3
2 log 27 x < _ 3
Original inequality _2
35. log 9 (3x  1) = log 9 (4x )
x < 27 3
Logarithmic to Exponential Inequality
36. log 2 (x 2  18) = log 2 (3x )
x<9
Simplify.
37. log 3 (3x + 4) ≤ log 3 (x  2)
Example 5
38. EARTHQUAKE The magnitude of an earthquake is measured on a logarithmic scale called the Richter scale. The magnitude M is given by M = log 10 x, where x represents the amplitude of the seismic wave causing ground motion. How many times as great is the amplitude caused by an earthquake with a Richter scale rating of 10 as an aftershock with a Richter scale rating of 7?
Solve log 5 (p 2  2) = log 5 p. log 5 (p 2  2) = log 5 p
Original equation
2
p 2=p
Property of Equality
p2  p  2 = 0
Subtract p from each side.
(p  2)(p + 1) = 0 p2=0
Factor.
p+1=0
or
p=2
p = 1
Zero Product Property Solve each equation.
The solution is p = 2, since log 5 p is undefined for p = 1.
✔3103.3.16
855 Properties of Logarithms
(pp. 509–515)
Use log 5 16 ≈ 1.7227 and log 5 2 ≈ 0.4307 to approximate the value of each expression. 39. log 5 8
40. log 5 64
41. log 5 4
1 42. log 5 _
1 43. log 5 _
8
2
Solve each equation. Check your solution. 44. log 5 x  log 5 2 = log 5 15 45. 3 log 4 a = log 4 27 46. 2 log 3 x + log 3 3 = log 3 36 47. log 4 n + log 4 (n  4) = log 4 5 48. SOUND Use the formula L = 10 log 10 R, where L is the loudness of a sound and R is the sound’s relative intensity, to find out how much louder 20 people talking would be than one person talking. Suppose the sound of one person talking has a relative intensity of 80 decibels.
Example 6 Use log 5 16 ≈ 1.7227 and log 5 2 ≈ 0.4307 to approximate log 5 32. log 5 32 = log 5 (16 · 2)
Replace 32 with 16.
= log 5 16 + log 5 2
Product Property
≈ 1.7227 + 0.4307
Use a calculator.
≈ 2.1534
Example 7 Solve log 3 3x + log 3 4 = log 3 36. log 3 3x + log 3 4 = log 3 36
Original equation
log 3 3x (4) = log 3 36
Product Property
3x (4) = 36 12x = 36 x=3
Definition of logarithm Multiply. Divide each side by 12.
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Study Guide and Review Continued ✔3103.3.13, ✔3103.3.16
866 Common Logarithms
(pp. 516–522)
Example 8
Solve each equation or inequality. Round to the nearest tenthousandth. 49. 3 = 15 51. 8
m+1
Solve 5 3x > 7 x + 1.
x2
x
50. 6 = 28
= 30
52. 12
r1
5 3x > 7 x + 1 3x
= 7r
log 5 > log 7
54. 5 x + 2 ≤ 3 x
53. 3 5n > 24
Original inequality
x+1
3x log 5 > (x + 1) log 7
Property of Inequality Power Property
3x log 5 > x log 7 + log 7 Distributive Property
55. SAVINGS You deposited $1000 into an account that pays an annual interest rate r of 5% compounded quarterly. nt Use A = P (1 + _r ) .
n
3x log 5  x log 7 > log 7
Subtract x log 7.
x (3 log 5  log 7) > log 7
Distributive Property
a. How long will it take until you have $1500 in your account? b. How long it will take for your money to double?
log 7 x > __ 3 log 5  log 7
Divide by 3 log 5  log 7.
x > 0.6751
Use a calculator.
The solution set is {x  x > 0.6751}. ✔3103.3.13, ✔3103.3.16
877 Base e and Natural Logarithms
(pp. 525–531)
Example 9
Solve each equation or inequality. Round to the nearest tenthousandth. 56. 4e x  11 = 17
Solve 3e 5x + 1 = 10. Round to the nearest tenthousandth.
57. 2e x + 1 = 15
3e 5x + 1 = 10
x
58. ln 2x = 6
59. 2 + e > 9 5
60. ln (x + 3) < 5
61. e
x
Original equation
5x
Subtract 1 from each side.
5x
Divide each side by 3.
5x
Property of Equality
5x = ln 3
ln e x = x
3e = 9 e =3
> 18
62. SAVINGS If you deposit $2000 in an account paying 6.4% interest compounded continuously, how long will it take for your money to triple? Use A = Pe rt.
ln e = ln 3 ln 3 x=_ 5
Divide each side by 5.
x ≈ 0.2197
Use a calculator. ✔3103.3.13, ✔3103.3.16
888 Using Exponential and Logarithmic Functions
(pp. 533–539)
63. CARS Abe bought a used car for $2500. It is expected to depreciate at a rate of 25% per year. What will be the value of the car in 3 years? 64. BIOLOGY For a certain strain of bacteria, k is 0.728 when t is measured in days. Using the formula y = ae kt, how long will it take 10 bacteria to increase to 675 bacteria? 65. POPULATION The population of a city 20 years ago was 24,330. Since then, the population has increased at a steady rate each year. If the population is currently 55,250, find the annual rate of growth for this city.
Example 10 A certain culture of bacteria will grow from 250 to 2000 bacteria in 1.5 hours. Find the constant k for the growth formula. Use y = ae kt. y = ae kt 2000 = 250e 8 = e 1.5k ln 8 = ln e
1.5k
Replace y with 2000, a with 250, and t with 1.5. Divide each side by 250. Property of Equality
ln 8 = 1.5k
Inverse Property
ln 8 _ =k 1.5
Divide each side by 1.5.
1.3863 ≈ k
544  Chapter 8  Study Guide and Review
Exponential Growth Formula k (1.5)
Use a calculator.
Tennessee Curriculum Standards
Practice Test Graph each function. State the domain and range. 1. f(x) = 3 x  3 + 2 3 2. f(x) = 2 _
(4)
x+1
SPI 2.12, SPI 2.13, SPI 3103.3.10
1 17. MULTIPLE CHOICE What is the value of log 4 _ ? 64
A 3
3
Solve each equation or inequality. Round to four decimal places if necessary.
1 B _ 3
1 C _ 3
D 3
3. 8 c + 1 = 16 2c + 3 1 4. 9 x  2 > _
x
18. SAVINGS You put $7500 in a savings account paying 3% interest compounded continuously.
( 27 )
5. 2 a + 3 = 3 2a  1 2
6. log 2 (x  7) = log 2 6x 7. log 5 x > 2 8. log 3 x + log 3 (x  3) = log 3 4 9. 6 n  1 ≤ 11 n
a. Assuming there are no deposits or withdrawals from the account, what is the balance after 5 years? b. How long will it take your savings to double? c. In how many years will you have $10,000 in your account?
10. 4e 2x  1 = 5 11. ln (x + 2) 2 > 2
19. MULTIPLE CHOICE What is the solution of log 4 16  log 4 x = log 4 8?
Use log 5 11 ≈ 1.4899 and log 5 2 ≈ 0.4307 to approximate the value of each expression. 12. log 5 44
1 F _ 2
G 2 H 4
11 13. log 5 _ 2
J 8
14. POPULATION The population of a city 10 years ago was 150,000. Since then, the population has increased at a steady rate each year. The population is currently 185,000.
20. MULTIPLE CHOICE Which function is graphed below? y
a. Write an exponential function that could be used to model the population after x years if the population changes at the same rate. O
b. What will the population be in 25 years?
x
3 15. Write log 9 27 = _ in exponential form. 2
16. AGRICULTURE An equation that models the decline in the number of U.S. farms is y = 3,962,520(0.98) x, where x is the number of years since 1960 and y is the number of farms. a. How can you tell that the number is declining?
A y = log 10 (x  5) B y = 5 log 10 x C y = log 10 (x + 5) D y = 5 log 10 x
b. By what annual rate is the number declining? c. Predict when the number of farms will be less than 1 million.
1 21. Write 2 ln 6 + 3 ln 4  5 ln _ as a single logarithm.
(3)
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Preparing for Standardized Tests Using Technology Your calculator can be a useful tool in taking standardized tests. Some problems that you encounter might have steps or computations that require the use of a calculator. A calculator may also help you solve a problem more quickly.
Strategies for Using Technology Step 1 A calculator is a useful tool, but typically it should be used sparingly. Standardized tests are designed to measure your ability to reason and solve problems, not to measure your ability to punch keys on a calculator. Before using a calculator, ask yourself: • How would I normally solve this type of problem? • Are there any steps that I cannot perform mentally or by using paper and pencil? • Is a calculator absolutely necessary to solve this problem? • Would a calculator help me solve this problem more quickly or efficiently?
Step 2 When might a calculator come in handy? • solving problems that involve large, complex computations • solving certain problems that involve graphing functions, evaluating functions, solving equations, and so on • checking solutions of problems
SPI 3103.3.13
Test Practice Example Read the problem. Identify what you need to know. Then use the information in the problem to solve. A certain can of soda contains 60 milligrams of caffeine. The caffeine is eliminated from the body at a rate of 15% per hour. What is the halflife of the caffeine? That is, how many hours does it take for half of the caffeine to be eliminated from the body? A 4 hours
C 4.5 hours
B 4.25 hours
D 4.75 hours
546  Chapter 8  Preparing for Standardized Tests
Read the problem carefully. The problem can be solved using an exponential function. Use the exponential decay formula to model the problem and solve for the halflife of caffeine. y = a(1  r)t y = 60(1  0.15)t Half of 60 milligrams is 30. So, let y = 30 and solve for t. 30 = 60(1  0.15)t 0.5 = (0.85)t Take the log of each side and use the power property. log 0.5 = log (0.85)t log 0.5 = t log 0.85 log 0.5 _ =t log 0.85
At this point, it is necessary to use a calculator to evaluate the logarithms and solve the problem. Doing so shows that t ≈ 4.265. So, the halflife of caffeine is about 4.25 hours. The correct answer is B.
Exercises Read each problem. Identify what you need to know. Then use the information in the problem to solve. 1. Jason recently purchased a new truck for $34,750. The value of the truck decreases by 12% each year. What will the approximate value of the truck be 7 years after Jason purchased it? A $13,775 B $13,890
3. Lucinda deposited $2500 in a CD with the terms described below.
Super CD! Earn 4.25% interest compounded daily!
$$
(Minimum deposit of $1,000 over a period of at least 12 months.)
Use the formula below to solve for t, the number of years needed to earn $250 in interest with the CD.
C $14,125 D $14,200
0.0425 2750 = 2500 1 + _
(
2. A baseball is thrown upward at a velocity of 105 feet per second, and is released when it is 5 feet above the ground. The height of the baseball t seconds after being thrown is given by the formula h(t) = 16t2 + 105t + 5. Find the time at which the baseball reaches its maximum height. F 1.0 s
H 6.6 s
G 3.3 s
J
365
)
365t
A about 2.15 years B about 2.24 years C about 2.35 years D about 2.46 years
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Standardized Test Practice Cumulative, Chapters 1 through 8 6. Graph y = log5 x.
Multiple Choice
y
F Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper.
x
0
1. What is the yintercept of the exponential function below? y = 4x  1 A 0
B
1
C
2
D 3 y
G 2. Suppose there are only 3500 birds of a particular endangered species left on an island and the population decreases at a rate of about 5% each year. p The logarithmic function t = log0.95 _ predicts 3500 how many years t it will be for the population to decease to a number p. About how long will it take for the population to reach 3000 birds? F 2 years
H 5 years
G 3 years
J
3. Suppose a certain bacteria duplicates to reproduce itself every 20 minutes. If you begin with one cell of the bacteria, how many will there be after 2 hours? A 2
B
6
32
C
G 238
x
0
D 64
4. Lucas determined that the total cost C to rent a car could be represented by the equation C = 0.35m + 125, where m is the number of miles that he drives. If the total cost to rent the car was $363, how many miles did he drive? F 125
y
H
8 years
x
0
y
J
H 520 J
680
x 0
5. Which of the following best describes the graph of 3y = 4x  3 and 8y = 6x  5? A The lines have the same yintercept. B The lines have the same xintercept. C The lines are perpendicular. D The lines are parallel
7. Ray’s Book Store sells two used books for $7.99. The total cost c of purchasing n books can be found by— A multiplying n by c.
TestTakingTip
B multiplying n by 5.
Question 2 Use technology and the properties of logarithms to find t when p = 3000.
C multiplying n by the cost of 1 book. D dividing n by c
548  Chapter 8  Standardized Test Practice
13. GRIDDED RESPONSE For what value of x would the rectangle below have an area of 48 square units?
Short Response/Gridded Response
x
Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 1 8. The function y = _
x
(2)
x8
is graphed below. What is
the domain of the function? y
Extended Response x
0
Record your answers on a sheet of paper. Show your work. 14. Suppose the number of whitetail deer in a particular region has increased at an annual rate of about 10% since 1995. There were 135,000 deer in 1995.
9. Suzanne bought a new car this year for $33,750. The value of the car is expected to decrease by 10.5% per year. What will be the approximate value of the car 6 years after Suzanne purchased it? Show your work.
a. Write a function to model the number of whitetail deer after t years. b. About how many whitetail deer inhabited the region in 2000? Round your answer to the nearest hundred deer.
10. Figure QRST is shown on the coordinate plane. y
3 (3, 0)
2 (2, 0) 0
x
5 (2, 5)
4 (3, 5)
15. Sandy inherited $250,000 from her aunt in 1998. She invested the money and increased it as shown in the table below.
What transformation creates an image with a vertex at the origin? 11. GRIDDED RESPONSE If f(x) = 3x and g(x) = what is the value of f(g(3))?
x2
Year
Amount
1998
$250,000
2006
$329,202
2011
$390,989
a. Write an exponential function that could be used to predict the amount of money A after investing for t years.
 1,
b. If the money continues to grow at the same rate, in what year will it be worth $500,000?
12. Simplify (2a2b6)(3a1b8).
Need ExtraHelp? If you missed Question... Go to Lesson... For help with TN SPI...
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