EXPONENTIAL AND LOGARITHMIC...

� � � � � � � �EXPONENTIAL ANDLOGARITHMICFUNCTIONS

EXPONENTIAL ANDLOGARITHMICFUNCTIONS

c How does altitude affect the air?

462

� � � � � � � �

APPLICATION: Mountain Climbing

A surprising fact that you may

not know is that air has weight!

The weight of the air above you

produces what scientists call

atmospheric pressure.

Mountain climbers need to be

aware of changes in atmospheric

pressure because as the pressure

decreases, so does the amount of

oxygen they have to breathe.

Think & Discuss

The graph below shows the relationship between

atmospheric pressure and altitude.

1. Describe what happens to the atmospheric pressure

as the altitude increases.

2. Mount McKinley in Alaska is 20,320 feet high.

Estimate the atmospheric pressure at its peak.

Learn More About It

You will find the atmospheric pressure at the peak of

Mount Everest in Exercise 79 on p. 484.

APPLICATION LINK Visit www.mcdougallittell.com

for more information about atmospheric pressure.

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C H A P T E R

8

463

c

Variation of AtmosphericPressure with Altitude

Altitude (thousands of feet)

Pre

ss

ure

(lb

/in

.2)

10 20 30 40 500

0

4

8 �

� � � � � � � �

464 Chapter 8

What’s the chapter about?

Chapter 8 is about exponential and logarithmic functions. These functions areinverses of each other. In Chapter 8 you’ll learn

• how to graph and use exponential, logarithmic, and logistic growth functions.

• how to use the number e and the definition and properties of logarithms.

• how to solve exponential and logarithmic equations.

CHAPTER

8Study Guide

PREVIEW

Are you ready for the chapter?

SKILL REVIEW Do these exercises to review key skills that you’ll apply in thischapter. See the given reference page if there is something you don’t understand.

Evaluate the expression. (Review Example 1, p. 11; Example 1, p. 324)

1. 4º3 2. S}

13

}D23. S}

34

}D04. º52 5. S}

52

}Dº1

Describe the end behavior of the graph of the function by completing the

statements ƒ(x) ˘ ooo

? as x ˘ º‡ and ƒ(x) ˘ ooo

? as x ˘ +‡. (Review Example 4, p. 332)

6. ƒ(x) = 2x3 7. ƒ(x) = ºx

2 8. ƒ(x) = 4x4 9. ƒ(x) = º5x

3

Draw a scatter plot of the data. Then approximate an equation of the best-

fitting line. (Review Example 2, p. 101)

10.

PREPARE

Here’s a study strategy!

c Review

• base, p. 11

• inverse function, p. 422

c New

• exponential function, p. 465

• asymptote, p. 465

• exponential growth function, p. 466

• exponential decay function, p. 474

• natural base e, p. 480

• logarithm of y with base b, p. 486

• common logarithm, p. 487

• natural logarithm, p. 487

• logistic growth function, p. 517

STUDENT HELP

Study Tip“Student Help” boxesthroughout the chaptergive you study tips andtell you where to look for extra help in this book and on the Internet.

Study Group

Form a study group. Have each group member takelessons from the chapter and summarize theimportant concepts and skills in those lessons.Then have each member lead a discussion on howto solve the types of problems in his or her lessons.

x 1 2 3 4 5 6 7 8 9 10

y 2.2 2.9 3.0 4.1 4.2 4.3 4.8 5.0 5.9 5.9

STUDY

STRATEGY

KEY VOCABULARY

� � � � � �Exponential Growth

GRAPHING EXPONENTIAL GROWTH FUNCTIONS

An involves the expression bx where the b is a positivenumber other than 1. In this lesson you will study exponential functions for whichb > 1. To see the basic shape of the graph of an exponential function such asƒ(x) = 2x, you can make a table of values and plot points, as shown below.

Notice the end behavior of the graph. As x ˘ +‡, ƒ(x) ˘ +‡, which means that thegraph moves up to the right. As x ˘ º‡, ƒ(x) ˘ 0, which means that the graph hasthe line y = 0 as an asymptote. An is a line that a graph approaches asyou move away from the origin.

In the activity you may have observed the following about the graph of y = a • 2x:

• The graph passes through the point (0, a). That is, the y-intercept is a.

• The x-axis is an asymptote of the graph.

• The domain is all real numbers.

• The range is y > 0 if a > 0 and y < 0 if a < 0.

asymptote

baseexponential function

GOAL 1

8.1 Exponential Growth 465

Graph exponential

growth functions.

Use exponential

growth functions to model

real-life situations, such as

Internet growth in Example 3.

. To solve real-life

problems, such as finding the

amount of energy generated

from wind turbines in

Exs. 49–51.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

8.1R

EAL LIFE

REA

L LIFE

Investigating Graphs of Exponential Functions

Graph y = }

13

} • 2x and y = 3 • 2x. Compare the graphs with the graph of y = 2x.

Graph y = º}

15

} • 2x and y = º5 • 2x. Compare the graphs with the graphof y = 2x.

Describe the effect of a on the graph of y = a • 2x when a is positive andwhen a is negative.

3

2

1

DevelopingConcepts

ACTIVITY

��

(1, 2)

(0, 1)

(2, 4)

(3, 8)

f (x) 5 2x

12s21, d14s22, d

18s23, d

x ƒ(x) = 2 x

º3 2º3 = }

81

}

º2 2º2 = }

14

}

º1 2º1 = }

12

}

0 20 = 1

1 21 = 2

2 22 = 4

3 23 = 8

� � � � � � � �

466 Chapter 8 Exponential and Logarithmic Functions

The characteristics of the graph of y = a • 2x listed on the previous page are true of the graph of y = abx. If a > 0 and b > 1, the function y = abx is an

Graphing Exponential Functions of the Form y = abx

Graph the function.

a. y = }

12

} • 3x b. y = ºS}

32

}Dx

SOLUTION

a. Plot S0, }

12

}D and S1, }

32

}D. Then,

from left to right, draw a curve that begins just above the x-axis, passes through the two points, and movesup to the right.

. . . . . . . . . .

To graph a general exponential function,

y = abx º h + k,

begin by sketching the graph of y = abx. Then translate the graph horizontally by hunits and vertically by k units.

Graphing a General Exponential Function

Graph y = 3 • 2x º 1 º 4. State the domain and range.

SOLUTION

Begin by lightly sketching the graph of y = 3 • 2x, which passes through (0, 3)and (1, 6). Then translate the graph 1 unit to the right and 4 units down.Notice that the graph passes through (1, º1) and (2, 2). The graph’sasymptote is the line y = º4. Thedomain is all real numbers, and therange is y > º4.

E X A M P L E 2

E X A M P L E 1

exponential growth function.

��

(1, 21)

y 5 3 p 2x ! (2, 2)

(0, 3)

(1, 6)

y 5 3 p 2x 2 1 2 4

��!" y 5 p 3x1

2

s1, d32s0, d1

2

� �!(0, 21)

2

" 32s1, 2 d

y 5 2x cx32

b. Plot (0, º1) and S1, º}

32

}D. Then,

from left to right, draw a curve thatbegins just below the x-axis, passesthrough the two points, and movesdown to the right.

Look Back For help with endbehavior of graphs, see p. 331.

STUDENT HELP

� � � � � � � �


USING EXPONENTIAL GROWTH MODELS

When a real-life quantity increases by a fixed percent each year (or other timeperiod), the amount y of the quantity after t years can be modeled by this equation:

y = a(1 + r)t

In this model, a is the initial amount and r is the percent increase expressed as adecimal. The quantity 1 + r is called the

Modeling Exponential Growth

INTERNET HOSTS In January, 1993, there were about 1,313,000 Internet hosts.During the next five years, the number of hosts increased by about 100% per year.

c Source: Network Wizards

a. Write a model giving the number h (in millions) of hosts t years after 1993.About how many hosts were there in 1996?

b. Graph the model.

c. Use the graph to estimate the year when there were 30 million hosts.

SOLUTION

a. The initial amount is a = 1.313 and the percent increase is r = 1. So, theexponential growth model is:

h = a(1 + r)tWrite exponential growth model.

= 1.313(1 + 1)tSubstitute for a and r.

= 1.313 • 2tSimplify.

Using this model, you can estimate the number of hosts in 1996 (t = 3) to be h = 1.313 • 23 ≈ 10.5 million.

b. The graph passes through the points (0, 1.313) and (1, 2.626). It has the t-axisas an asymptote. To make an accurategraph, plot a few other points. Then draw a smooth curve through the points.

c. Using the graph, you can estimate that thenumber of hosts was 30 million sometimeduring 1997 (t ≈ 4.5).

. . . . . . . . . .

In Example 3 notice that the annual percent increase was 100%. This translated into agrowth factor of 2, which means that the number of Internet hosts doubled each year.

People often confuse percent increase and growth factor, especially when a percentincrease is 100% or more. For example, a percent increase of 200% means that aquantity tripled, because the growth factor is 1 + 2 = 3. When you hear or readreports of how a quantity has changed, be sure to pay attention to whether a percentincrease or a growth factor is being discussed.

E X A M P L E 3

growth factor.

GOAL 2

#$

Years since 1993

Nu

mb

er

of

ho

sts

(mil

lio

ns)

0 1 2 3 4 5 60

Internet Hosts

30

20

10

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor extra examples.

INT

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STUDENT HELP

INTERNET HOSTS

A host is a computerthat stores information youcan access through theInternet. For example, Websites are stored on hostcomputers.

APPLICATION LINK

www.mcdougallittell.com

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% & ' ( ) * + ,


COMPOUND INTEREST Exponential growth functions are used in real-lifesituations involving compound interest. Compound interest is interest paid on theinitial investment, called the principal, and on previously earned interest. (Interestpaid only on the principal is called simple interest.)

Although interest earned is expressed as an annual percent, the interest is usuallycompounded more frequently than once per year. Therefore, the formula y = a(1 + r)t must be modified for compound interest problems.

Finding the Balance in an Account

FINANCE You deposit $1000 in an account that pays 8% annual interest. Find thebalance after 1 year if the interest is compounded with the given frequency.

a. annually b. quarterly c. daily

SOLUTION

a. With interest compounded annually, the balance at the end of 1 year is:

A = 1000S1 + }

0.108}D1 • 1

P = 1000, r = 0.08, n = 1, t = 1

= 1000(1.08)1 Simplify.

= 1080 Use a calculator.

c The balance at the end of 1 year is $1080.

b. With interest compounded quarterly, the balance at the end of 1 year is:

A = 1000S1 + }

0.408}D4 • 1

P = 1000, r = 0.08, n = 4, t = 1

= 1000(1.02)4 Simplify.

≈ 1082.43 Use a calculator.

c The balance at the end of 1 year is $1082.43.

c. With interest compounded daily, the balance at the end of 1 year is:

A = 1000S1 + }

03.6058

}D365 • 1P = 1000, r = 0.08, n = 365, t = 1

≈ 1000(1.000219)365 Simplify.

≈ 1083.28 Use a calculator.

c The balance at the end of 1 year is $1083.28.

E X A M P L E 4

Consider an initial principal P deposited in an account that pays interest at an

annual rate r (expressed as a decimal), compounded n times per year. The

amount A in the account after t years can be modeled by this equation:

A = PS1 + }

n

r}Dnt

COMPOUND INTEREST

FINANCIAL

PLANNER

Financial planners interviewclients to determine theirassets, liabilities, and finan-cial objectives. They analyzethis information and developan individual financial plan.

CAREER LINK


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FOCUS ON

CAREERS

% & ' ( ) * + ,


1. What is an asymptote?

2. Given the general exponential function ƒ(x) = abx º h + k, describe the effects of a, h, and k on the graph of the function.

3. For what values of b does y = bx represent exponential growth?

Graph the function. State the domain and range.

4. y = 4x 5. y = 3x º 1 6. y = 2x + 2

7. y = 5x º 3 8. y = 5x + 1 + 2 9. y = 2x º 3 + 1

10. What is the asymptote of the graph of y = 3 • 4x º 1 + 2? What is the value of ywhen x = 2?

11. POPULATION The population of Winnemucca, Nevada, can be modeled byP = 6191(1.04)t where t is the number of years since 1990. What was thepopulation in 1990? By what percent did the population increase each year?

12. ACCOUNT BALANCE You deposit $500 in an account that pays 3% annualinterest. Find the balance after 2 years if the interest is compounded with thegiven frequency.

a. annually b. quarterly c. daily

INVESTIGATING GRAPHS Identify the y-intercept and the asymptote of the

graph of the function.

13. y = 5x 14. y = º2 • 4x 15. y = 4 • 2x

16. y = 2x º 1 17. y = 3 • 2x º 1 18. y = 2 • 3x º 4

MATCHING GRAPHS Match the function with its graph.

19. y = 2 • 5x 20. y = 3 • 4x 21. y = º2 • 5x

22. y = }

13

} • 4x 23. y = 3x º 2 24. y = 3x º 2

A. B. C.

D. E. F. -. /- (3, 3)

(2, 1)

34s21, d -. /0

(0, 3)1. /- (1, 1)

(0, 21)

-. /2(0, 2)2

5s21, d(0, 22)

. /-2

225s21, 2 d-. /2

43s1, d1

3s0, d

PRACTICE AND APPLICATIONS

GUIDED PRACTICE

Vocabulary Check ✓

Skill Check ✓

Concept Check ✓

Extra Practiceto help you masterskills is on p. 950.

STUDENT HELP

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 13–15, 19–22, 25–33

Example 2: Exs. 16–18,23, 24, 34–42

Example 3: Exs. 43–54,56, 58, 66

Example 4: Exs. 55, 57,59–65, 67

3 4 5 6 7 8 9 :


GRAPHING FUNCTIONS Graph the function.

25. y = 5x 26. y = º2x 27. y = 8 • 2x

28. y = º3 • 2x 29. y = º2 • 5x 30. y = º(2.5)x

31. y = 6S}

54

}Dx

32. y = º}

23

} • 3x 33. y = º}

15

}(1.5)x

GRAPHING FUNCTIONS Graph the function. State the domain and range.

34. y = º2 • 3x + 2 35. y = 4 • 5x º 1 36. y = 7 • 3x º 2

37. y = 3 • 4x º 1 38. y = 3x + 1 + 1 39. y = 2x º 3 + 3

40. y = º3 • 6x + 2 º 2 41. y = 4 • 2x º 3 + 1 42. y = 8 • 2x º 3 º 3

NATURAL GAS In Exercises 43–45, use the following information.

The amount g (in trillions of cubic feet) of natural gas consumed in the United Statesfrom 1940 to 1970 can be modeled by

g = 2.91(1.07)t

where t is the number of years since 1940. c Source: Wind Energy Comes of Age

43. Identify the initial amount, the growth factor, and the annual percent increase.

44. Graph the function.

45. Estimate the natural gas consumption in 1955.

COMPUTER CHIPS In Exercises 46–48, use the following information.

From 1971 to 1995, the average number n of transistors on a computer chip can bemodeled by

n = 2300(1.59)t

where t is the number of years since 1971.

46. Identify the initial amount, the growth factor, and the annual percent increase.

47. Graph the function.

48. Estimate the number of transistors on a computer chip in 1998.

WIND ENERGY In Exercises 49–51, use the following information.

In 1980 wind turbines in Europe generated about 5 gigawatt-hours of energy. Overthe next 15 years, the amount of energy increased by about 59% per year.

49. Write a model giving the amount E (in gigawatt-hours) of energy t years after1980. About how much wind energy was generated in 1984?

50. Graph the model.

51. Estimate the year when 80 gigawatt-hours of energy were generated.

FEDERAL DEBT In Exercises 52–54, use the following information.

In 1965 the federal debt of the United States was $322.3 billion. During the next30 years, the debt increased by about 10.2% each year. c Source: U.S. Bureau of the Census

52. Write a model giving the amount D (in billions of dollars) of debt t years after1965. About how much was the federal debt in 1980?


54. Estimate the year when the federal debt was $2,120 billion.

FEDERAL DEBT

When the govern-ment has an annual deficit it must borrow money. Theaccumulation of thisborrowing is called theFederal debt.

RE

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RE

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FOCUS ON

APPLICATIONS

0

1

2

3

4

5

1965 1975 1985 1995

De

bt

(tri

llio

ns

of

do

lla

rs)

Federal Debt

; < = > ? @ A B


55. EARNING INTEREST You deposit $2500 in a bank that pays 4% interestcompounded annually. Use the process below and a graphing calculator to

determine the balance of your account each year.

a. Enter the initial deposit, 2500, into the calculator. Then enter the formulaANS + ANS ª 0.04 to find the balance after one year.

b. What is the balance after five years? (Hint: The balance after each year will bedisplayed each time you press the key.)

c. How would you enter the formula in part (a) if the interest is compoundedquarterly? What do you have to do to find the balance after one year?

d. Find the balance after 5 years if the interest is compounded quarterly.Compare this result with your answer to part (b).

WRITING MODELS In Exercises 56–58, write an exponential growth model

that describes the situation.

56. COIN COLLECTING You buy a commemorative coin for $110. Each year t,the value V of the coin increases by 4%.

57. SAVINGS ACCOUNT You deposit $400 in an account that pays 2% annualinterest compounded quarterly.

58. ANTIQUES You purchase an antique table for $525. Each year t, the value Vof the table increases by 5%.

ACCOUNT BALANCE In Exercises 59–61, use the following information.

You deposit $1600 in a bank account. Find the balance after 3 years for each of thefollowing situations.

59. The account pays 2.5% annual interest compounded monthly.

60. The account pays 1.75% annual interest compounded quarterly.

61. The account pays 4% annual interest compounded yearly.

DEPOSITING FUNDS In Exercises 62–64, use the following information.

You want to have $2500 after 2 years. Find the amount you should deposit for eachof the situations described below.

62. The account pays 2.25% annual interest compounded monthly.

63. The account pays 2% annual interest compounded quarterly.

64. The account pays 5% annual interest compounded yearly.

65. CRITICAL THINKING Juan and Michelle each have $800. Juan plans to invest$200 for each of the next four years, while Michelle plans to invest all $800 now.Both accounts pay 3% annual interest compounded monthly. Will they have thesame amount of money after four years? If not, explain why.

66. LAND VALUE You have inherited land that was purchased for $30,000 in1960. The value V of the land increased by approximately 5% per year.

a. Write a model for the value of the land t years after 1960.

b. What is the approximate value of the land in the year 2010?

67. LOGICAL REASONING Is investing $4000 at 5% annual interest and $4000 at7% annual interest equivalent to investing $8000 (the total of the two principals)at 6% annual interest (the average of the two interest rates)? Explain.

KEYSTROKE HELP

Visit our Web sitewww.mcdougallittell.comto see keystrokes for several models ofcalculators.

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; < = > ? @ A B


68. MULTIPLE CHOICE The student enrollment E of a high school was 1240 in1990 and increased by 15% per year until 1996. Which exponential growthmodel shows the school’s student enrollment in terms of t, the number of yearssince 1990?

¡A E = 15(1240)t¡B E = 1240(1.15)t

¡C E = 1240(15)t

¡D E = 0.15(1240)t¡E E = 1.15(1240)t

69. MULTIPLE CHOICE Which function is graphed below?

¡A ƒ(x) = º3x + 1 + 6

¡B ƒ(x) = º2 • 3x + 1 + 6

¡C ƒ(x) = 2 • 3x + 1 + 6

¡D ƒ(x) = 2 • 3x º 1 + 6

¡E ƒ(x) = 3x + 1 + 6

70. IRRATIONAL EXPONENTS Use a calculator to evaluate the followingpowers. Round the results to five decimal places.

314/10, 3141/100, 31,414/1,000, 314,142/10,000, 3141,421/100,000, 31,414,213/1,000,000

Each of these powers has a rational exponent. Explain how you can use these

powers to define 3Ï2w, which has an irrational exponent.

EVALUATING POWERS Evaluate the expression. (Review 1.2 for 8.2)

71. S}

12

}D372. S}

37

}D373. S}

12

}D574. S}

58

}D4

75. S}

172}D3

76. S}

23

}D477. S}

45

}D278. S}

130}D5

EVALUATING EXPRESSIONS Evaluate the expression using a calculator.

Round the result to two decimal places when appropriate. (Review 7.1)

79. 83/8 80. 15,6251/6 81. º2431/5 82. 10241/5

83. 101/2 84. 1061/3 85. Ï48w1w 86. Ï7

1w0w0w

87. Ï32w8w 88. Ï4

1w2w0w 89. Ï49w 90. Ï6

1w8w0w

OPERATIONS WITH FUNCTIONS Let ƒ(x) = 6x º 11 and g(x) = 4x2. Perform

the indicated operation and state the domain. (Review 7.3)

91. ƒ(x) + g(x) 92. ƒ(x) º g(x) 93. ƒ(x) • g(x)

94. g(x) º ƒ(x) 95. ƒ(g(x)) 96. g(ƒ(x))

97. }

ƒ

g

(

(

x

x

)

)} 98. }

g

ƒ

(

(

x

x

)

)} 99. ƒ(ƒ(x))

100. FENCING You want to build a rectangular pen for your dog using 40 feetof fencing. The area of the pen should be 90 square feet. What should thedimensions of the pen be? (Review 5.2)

MIXED REVIEW

TestPreparation

★★Challenge

CD

EC (1, 8)

23s0, 6 d

EXTRA CHALLENGE


; < = > ? @ A BDeveloping Concepts

ACTIVITY 8.2

SET UP

Work with a partner.

MATERIALS

• paper

• pencil

• graph paper

8.2 Concept Activity 473

Group Activity for use with Lesson 8.2

Fold number 0 1 2 3 4 5

Number of regions 1 2 ? ? ? ?

Fractional area of each region 1 }

12

} ? ? ? ?

Exponential Growth and Decay

c QUESTION What relationships exist between exponential growth and

exponential decay when a piece of paper is folded repeatedly?

c EXPLORING THE CONCEPT

Fold a rectangular piece of paper in half. The fold divides the paper into two regions,

each of which has half the area of the paper.

Fold the paper in half again. Into how many regions has the original piece of paper

been folded? What fraction of the paper’s area does each region have?

Continue to fold the paper until it is no longer possible to make another fold. After

each fold, record in a table like the one shown the fold number, the number of regions

into which the paper has been folded, and the fraction of the paper’s area that each

region has.

Make two scatter plots of the data in the table. The first scatter plot will have ordered

pairs of the form ( fold number, number of regions) and the second will have ordered

pairs of the form ( fold number, fractional area of each region).

c DRAWING CONCLUSIONS

1. The first scatter plot is an example of exponential growth. Write an equation forthe graph.

2. Use the equation from Exercise 1 to determine the number of regions therewould be after 8 folds.

3. The second scatter plot is an example of exponential decay. Write an equation forthe graph.

4. Use the equation from Exercise 3 to determine the fractional area of each regionafter 8 folds.

5. Multiply the exponential expressions from Exercise 2 and Exercise 4. Explainwhy the product should be 1.

4

3

2

1

F G H I J K L MExponential Decay

GRAPHING EXPONENTIAL DECAY FUNCTIONS

In Lesson 8.1 you studied exponential growth functions. In this lesson you will studywhich have the form ƒ(x) = abx where a > 0 and

0 < b < 1.

Recognizing Exponential Growth and Decay

State whether ƒ(x) is an exponential growth or exponential decay function.

a. ƒ(x) = 5S}

23

}Dxb. ƒ(x) = 8S}

32

}Dxc. ƒ(x) = 10(3)ºx

SOLUTION

a. Because 0 < b < 1, ƒ is an exponential decay function.

b. Because b > 1, ƒ is an exponential growth function.

c. Rewrite the function as ƒ(x) = 10S}

13

}Dx. Because 0 < b < 1, ƒ is an exponential

decay function.

. . . . . . . . . .

To see the basic shape of the graph of an exponential decay function, you can make atable of values and plot points, as shown below.

Notice the end behavior of the graph. As x ˘ º‡, ƒ(x) ˘ +‡, which means that thegraph moves up to the left. As x ˘ +‡, ƒ(x) ˘ 0, which means that the graph hasthe line y = 0 as an asymptote.

E X A M P L E 1

exponential decay functions,

GOAL 1


Graph exponential

decay functions.

Use exponential

decay functions to model

real-life situations, such as

the decline of record sales in

Exs. 47–49.



depreciated value of a car in

Example 4.


GOAL 2

GOAL 1


8.2R

EAL LIFE

REA

L LIFEx ƒ(x) = S}

12

}Dx

º3 S}

12

}Dº3= 8

º2 S}

12

}Dº2= 4

º1 S}

12

}Dº1= 2

0 S}

12

}D0= 1

1 S}

12

}D1= }

21

}

2 S}

12

}D2= }

14

}

3 S}

12

}D3= }

18

}

N OP

(23, 8)

(22, 4)

(21, 2)

(0, 1) Qf (x) 5 x cx12

12s1, d

14s2, d

18s3, d

R S T U V W X Y

8.2 Exponential Decay 475

Recall that in general the graph of an exponential function y = abx passes throughthe point (0, a) and has the x-axis as an asymptote. The domain is all real numbers,and the range is y > 0 if a > 0 and y < 0 if a < 0.

Graphing Exponential Functions of the Form y = abx

Graph the function.

a. y = 3S}

14

}Dxb. y = º5S}

23

}Dx

SOLUTION

a. Plot (0, 3) and S1, }

34

}D. b. Plot (0, º5) and S1, º}

130}D.

Then, from right to left, draw Then, from right to left, drawa curve that begins just above a curve that begins just belowthe x-axis, passes through the the x-axis, passes through thetwo points, and moves up two points, and moves down to the left. to the left.

. . . . . . . . . .

Remember that to graph a general exponential function, y = abx º h + k, begin bysketching the graph of y = abx. Then translate the graph horizontally by h units andvertically by k units.

Graphing a General Exponential Function

Graph y = º3S}

12

}Dx + 2+ 1. State the domain and range.

SOLUTION

Begin by lightly sketching the graph

of y = º3S}

12

}Dx, which passes through (0, º3)

and S1, º}

32

}D. Then translate the graph 2 units

to the left and 1 unit up. Notice that the graph

passes through (º2, º2) and Sº1, º}

12

}D. The

graph’s asymptote is the line y = 1. The domainis all real numbers, and the range is y < 1.

E X A M P L E 3

2

Q Q OP(0, 25)

y 5 25x cx23

103s1, 2 dQ O

PQ(0, 3)

34s1, d

y 5 3x cx14

E X A M P L E 2

QQ OP(0, 23)

(22, 22)

y 5 23x cx12

32s1, 2 d

y 5 23x cx 1 2 1 1

12

12s21, 2 d

R S T U V W X Y


USING EXPONENTIAL DECAY MODELS

When a real-life quantity decreases by a fixed percent each year (or other timeperiod), the amount y of the quantity after t years can be modeled by the equation

y = a(1 º r)t

where a is the initial amount and r is the percent decrease expressed as a decimal.The quantity 1 º r is called the

Modeling Exponential Decay

You buy a new car for $24,000. The value y of the car decreases by 16% each year.

a. Write an exponential decay model for the value of the car. Use the model toestimate the value after 2 years.

b. Graph the model.

c. Use the graph to estimate when the car will have a value of $12,000.

SOLUTION

a. Let t be the number of years since you bought the car. The exponential decaymodel is:

y = a(1 º r)tWrite exponential decay model.

= 24,000(1 º 0.16)tSubstitute for a and r.

= 24,000(0.84)tSimplify.

When t = 2, the value is y = 24,000(0.84)2 ≈ $16,934.

b. The graph of the model is shown at the right. Notice that it passesthrough the points (0, 24,000) and(1, 20,160). The asymptote of thegraph is the line y = 0.

c. Using the graph, you can see that thevalue of the car will drop to $12,000after about 4 years.

. . . . . . . . . .

In Example 4 notice that the percent decrease, 16%, tells you how much value thecar loses from one year to the next. The decay factor, 0.84, tells you what fraction ofthe car’s value remains from one year to the next. The closer the percent decrease forsome quantity is to 0%, the more the quantity is conserved or retained over time. Thecloser the percent decrease is to 100%, the more the quantity is used or lost over time.

E X A M P L E 4

decay factor.

GOAL 2

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Automobiles

Years since purchase

Va

lue

(d

oll

ars

)

Z0

24,000

20,000

16,000

12,000

8000

4000

01 2 3 4 5 6 7 8 9 [

Car Depreciation

HOMEWORK HELP


INT

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STUDENT HELP

\ ] ^ _ ` a b c


1. In the exponential decay model y = 1500(0.65)t, identify the initial amount, thedecay factor, and the percent decrease.

2. What is the asymptote of the graph of the function y = 2S}

15

}Dx º 2+ 3?

3. For what values of b does y = bx represent exponential decay?


4. y = º(0.5)x 5. y = 2S}

13

}Dx6. y = 4S}

23

}Dx

7. y = º5S}

23

}Dx º 28. y = º4(0.25)x + 1 9. y = 5S}

12

}Dx+ 2

10. RADIOACTIVE DECAY The amount y (in grams) of a sample of iodine-131after t days is given by y = 50(0.92)t.

a. Identify the initial amount of the substance.

b. What percent of the substance decays each day?

IDENTIFYING FUNCTIONS Tell whether the function represents exponential

growth or exponential decay.

11. ƒ(x) = 4S}

38

}Dx12. ƒ(x) = 10 • 3x 13. ƒ(x) = 8 • 7ºx 14. ƒ(x) = 8 • 7x

15. ƒ(x) = 5S}

18

}Dºx16. ƒ(x) = 3S}

43

}Dx17. ƒ(x) = 8S}

23

}Dx18. ƒ(x) = 5(0.25)ºx


19. y = (0.25)x 20. y = º3x º 1 + 3 21. y = ºS}

13

}Dx º 1+ 3

22. y = S}

12

}Dx º 123. y = º(0.25)x 24. y = (0.5)x º 1

A. B. C.

D. E. F. de fg(21, 4)

(0, 1)

d d fg(1, 2)

(2, 0)

h d fg(0, 0)

(1, 2)

2

d fg(0, 21)

(21, 24)dd fg(0, 2)

(1, 1)

d fg(0, 0)

12s1, 2 d


GUIDED PRACTICE


Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 11–18Example 2: Exs. 19, 23,

25–33Example 3: Exs. 20–22,

24, 34–42Example 4: Exs. 43–56


STUDENT HELP

\ ] ^ _ ` a b c


GRAPHING FUNCTIONS Graph the function.

25. y = 3S}

12

}Dx26. y = 2S}

15

}Dx27. y = º2S}

14

}Dx

28. y = º5S}

12

}Dx29. y = 4S}

13

}Dx30. y = 5S}

14

}Dx

31. y = º3S}

23

}Dx32. y = º5(0.75)x 33. y = 3S}

38

}Dx


34. y = ºS}

12

}Dx+ 1 35. y = S}

23

}Dx º 136. y = 4S}

12

}Dx + 1

37. y = S}

13

}Dx º 238. y = 2S}

13

}Dx º 139. y = (0.25)x + 3

40. y = º3S}

13

}Dx º 141. y = S}

13

}Dxº 2 42. y = S}

23

}Dxº 1

WRITING MODELS In Exercises 43–45, write an exponential decay model that

describes the situation.

43. STEREO SYSTEM You buy a stereo system for $780. Each year t, the valueV of the stereo system decreases by 5%.

44. BEVERAGES You drink a beverage with 120 milligrams of caffeine. Eachhour h, the amount c of caffeine in your system decreases by about 12%.

45. MEDICINE An adult takes 400 milligrams of ibuprofen. Each hour h, theamount i of ibuprofen in the person’s system decreases by about 29%.

46. RADIOACTIVE DECAY One hundred grams of plutonium is stored in acontainer. The amount P (in grams) of plutonium present after t years can bemodeled by this equation:

P = 100(0.99997)t

How much plutonium is present after 20,000 years?

RECORD ALBUMS In Exercises 47–49, use the following information.

The number A (in millions) of record albums sold each year in the United States from1982 to 1993 can be modeled by

A = 265(0.39)t

where t represents the number of years since 1982.

DATA UPDATE of Recording Industry Association of America data at www.mcdougallittell.com

47. Identify the initial amount, the decay factor, and the annual percent decrease.


49. Estimate when the number of records sold was 1 million.

DEPRECIATION In Exercises 50–52, use the following information.

You buy a new car for $22,000. The value of the car decreases by 12.5% each year.

50. Write an exponential decay model for the value of the car. Use the model toestimate the value after 3 years.


52. Estimate when the car will have a value of $8000.

INT

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\ ] ^ _ ` a b c


COMPUTERS In Exercises 53–55, use the following information.

You buy a new computer for $2100. The value of the computer decreases by about50% annually.

53. Write an exponential decay model for the value of the computer. Use the modelto estimate the value after 2 years.


55. Estimate when the computer will have a value of $600.

56. During normal breathing, about 12% of the air in thelungs is replaced after one breath. Write an exponential decay model for theamount of the original air left in the lungs if the initial amount of air in the lungsis 500 milliliters. How much of the original air is present after 240 breaths?

57. MULTI-STEP PROBLEM A new automobile worth $18,354 depreciates byabout 17% each year. The payoff amount on a loan after making n monthlypayments is given by the model

A(n) = SA0 º }}

Pr

}D(1 + r)n + }}

Pr

}

where A0 is the original amount of the loan, P is the monthly payment, and r isthe monthly interest rate expressed as a decimal.

a. Write an exponential decay model for the value V of the automobile t yearsafter it is purchased.

b. Write a model for the payoff amount on a loan of $18,354 with a monthlypayment of $280 and an annual interest rate of 8.5%. (Hint: The model for thepayoff amount uses the monthly interest rate, not the annual interest rate.)

c. Writing Make a table showing the value of the car and the payoff amounton the loan for 5 years. When would it make sense to sell the car? Explain.

58. CRITICAL THINKING Is the product of two exponential decay functions alwaysanother exponential decay function? Is the quotient of two exponential decayfunctions always another exponential decay function? Justify your answers.

GRAPHING FUNCTIONS Graph the function. (Review 7.5)

59. y = (x + 1)1/3 60. y = Ï3xw + 1 61. y = º3x1/3

62. y = Ïxw + 4 63. y = ºÏxw+w 5w 64. y = Ï3

xw + }

14

}

USING A DATA SET Find the mean, the median, the mode, and the range for

the set of data. (Review 7.7)

65. 11, 18, 13, 15, 17, 15, 23, 20, 12 66. 25, 30, 32, 42, 31, 33, 36, 22

67. FINANCE You deposit $2000 in a bank account. Find the balance after 4 years for each of the following situations. (Review 8.1 for 8.3)

a. The account pays 7% annual interest compounded quarterly.

b. The account pays 5% annual interest compounded monthly.

SCIENCE CONNECTION

TestPreparation

★★Challenge

MIXED REVIEW

i j k l m n o p


The Number e

USING THE NATURAL BASE e

The history of mathematics is marked by the discovery of special numbers such ascounting numbers, zero, negative numbers, π, and imaginary numbers. In this lessonyou will study one of the most famous numbers of modern times. Like π and i, thenumber e is denoted by a letter. The number is called the or the

after its discoverer, Leonhard Euler (1707–1783).

In the activity you may have discovered that as n gets larger and larger, the

expression S1 + }

1n

}Dngets closer and closer to 2.71828 . . . , which is the value of e.

Simplifying Natural Base Expressions

Simplify the expression.

a. e3 • e4 b. }

1

5

0

e

e2

3

} c. (3eº4x)2

SOLUTION

a. e3 • e4 = e3 + 4 b. }

1

5

0

e

e2

3

} = 2e3 º 2 c. (3eº4x)2 = 32e(º4x)(2)

= e7 = 2e = 9eº8x = }e

98x}

E X A M P L E 1

Euler number,natural base e,

GOAL 1

Use the number e

as the base of exponential

functions.

Use the natural

base e in real-life situations,

such as finding the air

pressure on Mount Everest

in Ex. 79.

. To solve real-lifeproblems, such as finding

the number of listed

endangered species in

Example 5.


GOAL 2

GOAL 1


8.3R

EAL LIFE

REA

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Investigating the Natural Base e

Copy the table and use a calculator to complete the table.

Do the values in the table appear to be approaching a fixed decimal

number? If so, what is the number rounded to three decimal places?

2

1

DevelopingConcepts

ACTIVITY

The natural base e is irrational. It is defined as follows:

As n approaches +‡, S1 + }

n

1}Dn

approaches e ≈ 2.718281828459.

THE NATURAL BASE e

The grizzly bear was first

listed as threatened in 1975

and remains an endangered

species today.

n 101 102 103 104 105 106

S1 + }n1}Dn

2.594 ? ? ? ? ?

q r s t u v w x

8.3 The Number e 481

Evaluating Natural Base Expressions

Use a calculator to evaluate the expression: a. e2 b. eº0.06

SOLUTION

EXPRESSION KEYSTROKES DISPLAY

a. e2 [ex] 2

b. eº0.06 [ex] .06

. . . . . . . . . .

A function of the form ƒ(x) = aerx is called a natural base exponential function. Ifa > 0 and r > 0, the function is an exponential growth function, and if a > 0 and r < 0, the function is an exponential decay function. The graphs of the basicfunctions y = ex and y = eºx are shown below.

Graphing Natural Base Functions


a. y = 2e0.75x b. y = eº0.5(x º 2) + 1

SOLUTION

a. Because a = 2 is positive and b. Because a = 1 is positive andr = 0.75 is positive, the function r = º0.5 is negative, the function is an exponential growth function. is an exponential decay function.Plot the points (0, 2) and (1, 4.23) Translate the graph of y = eº0.5x

and draw the curve. to the right 2 units and up 1 unit.

The domain is all real numbers, The domain is all real numbers, and the range is all positive and the range is y > 1.real numbers.

yz{|(21, 5.48)

(2, 2)

(0, 1)

(23, 4.48)

y 5 e20.5x

y 5 e20.5(x 2 2) 1 1

{y

z} (1, 4.23)

y 5 2e 0.75x

(0, 2)

E X A M P L E 3

{y

z}(0, 1) (1, 0.368)

y 5 e2xExponential

decay{y

z}(0, 1)

(1, 2.718)

y 5 e xExponential

growth

E X A M P L E 2

7.389056

0.941765

LEONHARD EULER

continued hismathematical researchdespite losing sight in oneeye in 1735. He publishedmore than 500 books andpapers during his lifetime.Euler’s use of e appearedin his book Mechanica,published in 1736.

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FOCUS ON

PEOPLE

q r s t u v w xUSING e IN REAL LIFE

In Lesson 8.1 you learned that the amount A in an account earning interestcompounded n times per year for t years is given by

A = PS1 + }

nr

}Dnt

where P is the principal and r is the annual interest rate expressed as a decimal. As napproaches positive infinity, the compound interest formula approximates thefollowing formula for continuously compounded interest:

A = Pert

Finding the Balance in an Account

You deposit $1000 in an account that pays 8% annual interest compoundedcontinuously. What is the balance after 1 year?

SOLUTION

Note that P = 1000, r = 0.08, and t = 1. So, the balance at the end of 1 year is:

A = Pert = 1000e0.08(1) ≈ $1083.29

In Example 4 of Lesson 8.1, you found that the balance from daily compounding is$1083.28. So, continuous compounding earned only an additional $.01.

Using an Exponential Model

ENDANGERED SPECIES Since 1972 the U.S. Fish and Wildlife Service has

kept a list of endangered species in the United States. For the years 1972–1998,

the number s of species on the list can be modeled by

s = 119.6e0.0917t

where t is the number of years since 1972.

a. What was the number of endangered species in 1972?

b. Graph the model.

c. Use the graph to estimate when the number of endangered species reached 1000.

SOLUTION

a. In 1972, when t = 0, the model gives:

s = 119.6e0 = 119.6

So, there were about 120 endangered species on the list in 1972.

b. The graph of the model is shown.

c. Use the Intersect feature to determine that s reaches 1000 when t ≈ 23,which is about 1995.

E X A M P L E 5

E X A M P L E 4

GOAL 2


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Finance

Intersection

X=23.158151 Y=1000

MARINE

BIOLOGIST

A marine biologist studiessalt-water plants andanimals. Those who work forthe U.S. Fish and WildlifeService help maintainpopulations of manatees,walruses, and otherendangered species.

CAREER LINK


INT

ERNET

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FOCUS ON

CAREERS

~ � � � � � � �


1. What is the Euler number? Give an approximation of the Euler number roundedto three decimal places.

2. Tell whether the function ƒ(x) = }

14

}e2x is an example of exponential growth orexponential decay. Explain.

3. Is it possible to express e as a ratio of two integers? Explain.


4. e2 • e6 5. eº2 • 3e7 6. (2e5x)2 7. (4eº2)3

8. S}

12

}eº2D4

9. Ï3w6we4wxw 10. 11. }

3

1

6

2

e

eº

4

2}

12. What is the horizontal asymptote of the graph of ƒ(x) = 2ex º 2?

Graph the function.

13. y = eº2x 14. y = }

12

}ex 15. y = }

18

}e2x

16. ENDANGERED SPECIES Use the model in Example 5 to estimate thenumber of endangered species in 1998.

SIMPLIFYING EXPRESSIONS Simplify the expression.

17. e2 • e4 18. eº3 • e5 19. (3eº3x)º1 20. (3e4x)2

21. 3eº2 • e6 22. S}

14

}eº2D323. ex • eº3x • e5 24. Ï4we2wxw

25. (100e0.5x)º2 26. ex • 4e2x + 1 27. }

2ee

x

} 28. }

5

e

e5x

x

}

29. Ï32w7we6wxw 30. (32eº4x)3 31. }

64ee

3x

} 32. Ï36w4we9wxw

EVALUATING EXPRESSIONS Use a calculator to evaluate the expression.

Round the result to three decimal places.

33. e3 34. eº2/3 35. e1.7 36. e1/2

37. eº1/4 38. e3.2 39. e8 40. eº3

41. eº4 42. 2e1/2 43. º4eº3 44. 0.5e3.2

45. º1.2e5 46. 0.02eº0.3 47. 225eº50 48. º8.95e1/5

GROWTH OR DECAY? Tell whether the function is an example of exponential

growth or exponential decay.

49. ƒ(x) = 5eº3x 50. ƒ(x) = }

18

}e5x 51. ƒ(x) = eº4x 52. ƒ(x) = }

16

}e2x

53. ƒ(x) = }

14

}e2x 54. ƒ(x) = eº8x 55. ƒ(x) = e3x 56. ƒ(x) = }

14

}eºx

57. ƒ(x) = eº6x 58. ƒ(x) = }

38

}e7x 59. ƒ(x) = eº9x 60. ƒ(x) = e8x


ex

}

e2x

GUIDED PRACTICE


Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 17–32Example 2: Exs. 33–48Example 3: Exs. 49–75Example 4: Exs. 76–78Example 5: Exs. 79, 80

STUDENT HELP

Extra Practice to help you masterskills is on p. 950.

~ � � � � � � �



61. y = 3e0.5x 62. y = }

13

}e0.5x 63. y = }

12

}eº(x º 1)

64. y = eºx + 1 65. y = 3eºx º 2 66. y = 3ex º 2

A. B. C.

D. E. F.


67. y = eºx 68. y = 4ex 69. y = }

13

}ex

70. y = 3e2x + 2 71. y = 1.5eº0.5x 72. y = 0.1e2x º 4

73. y = }

13

}ex º 2 º 1 74. y = }

43

}ex º 3 + 1 75. y = 0.5eº2(x º 1) º 2

76. CONTINUOUS COMPOUNDING You deposit $975 in an account that pays5.5% annual interest compounded continuously. What is the balance after 6 years?

77. COMPARING FORMULAS You deposit $2500 in an account that pays 6%annual interest. Use the formulas at the top of page 482 to calculate the accountbalance after one year when the interest is compounded annually, semiannually,quarterly, monthly, and continuously. What do you notice? Explain.

78. Writing Compare the effects of compounding interest continuously and compounding interest daily using the formulas A = Pert and

A = PS1 + }36

r5

}D365t.

79. MOUNT EVEREST The air pressure P at sea level is about 14.7 pounds persquare inch. As the altitude h (in feet above sea level) increases, the air pressuredecreases. The relationship between air pressure and altitude can be modeled by:

P = 14.7eº0.00004h

Mount Everest in Tibet and Nepal rises to a height of 29,028 feet above sea level.What is the air pressure at the peak of Mount Everest?

80. RATE OF HEALING The area of a wound decreases exponentially withtime. The area A of a wound after t days can be modeled by

A = A0eº0.05t

where A0 is the initial wound area. If the initial wound area is 4 squarecentimeters, how much of the wound area is present after 14 days?

� ��(21, 3.69)

s1, d12

�� (2, 0.91)

s0, d13�� (0, 1)

(2, 21.59)

� �� (0, 3)(21, 1.82)� ��(0, 2) (1, 1.37)

�(22, 21.59)

� ��(0, 1)�

MOUNT EVEREST

In 1953 Sir EdmundHillary of New Zealand andTenzing Norgay, a NepaleseSherpa tribesman, becamethe first people to reach thetop of Mount Everest.

RE

AL LIFE

RE

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FOCUS ON

PEOPLE

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor help with Exs. 67–75.

INT

ERNET

STUDENT HELP

� � � � � � � �


★★Challenge

TestPreparation

81. MULTIPLE CHOICE What is the simplified form of !3§?

¡A 6e2Ï3xw ¡B 6xÏ3

e6w ¡C 6Ï3e1w6xw ¡D }

6xe2

} ¡E 6e2x

82. MULTIPLE CHOICE Which function is graphed at the right?

¡A ƒ(x) = 3ex º 2¡B ƒ(x) = 3ex º 2

¡C ƒ(x) = 3eºx º 2 ¡D ƒ(x) = 3eº(x + 2)

¡E ƒ(x) = 3ex + 2

83. CRITICAL THINKING Find a value of n for which S1 + }

1n

}Dngives the

value of e correct to 9 decimal places. Explain the process you used to find your answer.

FINDING INVERSE FUNCTIONS Find an equation for the inverse of the

function. (Review 7.4 for 8.4)

84. ƒ(x) = º3x 85. ƒ(x) = 6x + 7 86. ƒ(x) = º5x º 24

87. ƒ(x) = }

12

}x º 10 88. ƒ(x) = º14x + 7 89. ƒ(x) = º}

15

}x º 13

SOLVING EQUATIONS Solve the equation. (Review 7.6)

90. Ïxw = 20 91. Ï35wxwºw 4w + 7 = 10 92. 2(x + 4)2/3 = 8

93. Ïx2w ºw 4w = x º 2 94. Ïxw+w 3w = Ï2wxwºw 1w 95. Ï3wxwºw 5w º 3Ïxw = 0

Graph the function. State the domain and range. (Lessons 8.1, 8.2)

1. y = 4x º 1 2. y = 3x + 1 + 2 3. y = }

12

} • 5x º 1

4. y = º2S}

16

}Dx5. y = S}

58

}Dx+ 2 6. y = º2 • 6x º 3 + 3

Simplify the expression. (Lesson 8.3)

7. 2e3 • e4 8. 4eº5 • e7 9. (º3e2x)2 10. (5eº3)º4x

11. 12. 13. Ï1w6we2wxw 14. Ï31w2w5we6wxw

15. Graph the function ƒ(x) = º4e2x. (Lesson 8.3)

16. RADIOACTIVE DECAY One hundred grams of radium is stored in acontainer. The amount R (in grams) of radium present after t years can bemodeled by R = 100eº0.00043t. Graph the model. How much of the radium is present after 10,000 years? (Lesson 8.3)

6ex

}

e5x

3ex

}

4e

QUIZ 1 Self-Test for Lessons 8.1–8.3

MIXED REVIEW

8(81e11x)}

3e5xº2

��(24, 21.95)

(0, 1)

� � � � � � � �


Evaluate

logarithmic functions.

Graph logarithmic

functions, as applied in

Example 8.

. To model real-lifesituations, such as the

slope of a beach in

Example 4.


GOAL 2

GOAL 1


8.4R

EAL LIFE

REA

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Logarithmic Functions

EVALUATING LOGARITHMIC FUNCTIONS

You know that 22 = 4 and 23 = 8. However, for what value of x does 2x = 6?Because 22 < 6 < 23, you would expect x to be between 2 and 3. To find the exact x-value, mathematicians defined logarithms. In terms of a logarithm, x = log2 6 ≈ 2.585. (In the next lesson you will see how this x-value is obtained.)

This definition tells you that the equations logb y = x and bx = y are equivalent. Thefirst is in logarithmic form and the second is in exponential form. Given an equationin one of these forms, you can always rewrite it in the other form.

Rewriting Logarithmic Equations

LOGARITHMIC FORM EXPONENTIAL FORM

a. log2 32 = 5 25 = 32

b. log5 1 = 0 50 = 1

c. log10 10 = 1 101 = 10

d. log10 0.1 = º1 10º1 = 0.1

e. log1/2 2 = º1 S}

12

}Dº1= 2

. . . . . . . . . .

Parts (b) and (c) of Example 1 illustrate two special logarithm values that you shouldlearn to recognize.

E X A M P L E 1

GOAL 1

Let b and y be positive numbers, b ≠ 1. The is

denoted by logb y and is defined as follows:

logb y = x if and only if bx = y

The expression logb y is read as “log base b of y.”

logarithm of y with base b

DEFINITION OF LOGARITHM WITH BASE b

Let b be a positive real number such that b ≠ 1.

LOGARITHM OF 1 logb 1 = 0 because b0 = 1.

LOGARITHM OF BASE b logb b = 1 because b1 = b.

SPECIAL LOGARITHM VALUES

� � ¡ ¢ £ ¤ ¥

8.4 Logarithmic Functions 487

Evaluating Logarithmic Expressions

Evaluate the expression.

a. log3 81 b. log5 0.04 c. log1/2 8 d. log9 3

SOLUTION

To help you find the value of logb y, ask yourself what power of b gives you y.

a. 3 to what power gives 81? b. 5 to what power gives 0.04?

34 = 81, so log3 81 = 4. 5º2 = 0.04, so log5 0.04 = º2.

c. }

12

} to what power gives 8? d. 9 to what power gives 3?

S}

12

}Dº3= 8, so log1/2 8 = º3. 91/2 = 3, so log9 3 = }

12

}.

. . . . . . . . . .

The logarithm with base 10 is called the It is denoted by log10

or simply by log. The logarithm with base e is called the It canbe denoted by loge, but it is more often denoted by ln.

COMMON LOGARITHM NATURAL LOGARITHM

log10 x = log x loge x = ln x

Most calculators have keys for evaluating common and natural logarithms.

Evaluating Common and Natural Logarithms

EXPRESSION KEYSTROKES DISPLAY

a. log 5 5

b. ln 0.1 .1

Evaluating a Logarithmic Function

SCIENCE CONNECTION The slope s of a beach is related to the average diameter d(in millimeters) of the sand particles on the beach by this equation:

s = 0.159 + 0.118 log d

Find the slope of a beach if the average diameter of the sand particles is 0.25 millimeter.

SOLUTION

If d = 0.25, then the slope of the beach is:

s = 0.159 + 0.118 log 0.25 Substitute 0.25 for d.

≈ 0.159 + 0.118(º0.602) Use a calculator.

≈ 0.09 Simplify.

c The slope of the beach is about 0.09. This is a gentle slope that indicates a rise ofonly 9 meters for a run of 100 meters.

E X A M P L E 4

E X A M P L E 3

natural logarithm.common logarithm.

E X A M P L E 2

0.698970

–2.302585

SAND The tablegives the diameters

of different types of sand.Notice that the diameter of apebble is about 64 timeslarger than the diameter ofvery fine sand.

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Sand particle

Diameter(mm)

Pebble 4

Granule 2

Very coarse sand 1

Coarse sand 0.5

Medium sand 0.25

Fine sand 0.125

Very fine sand 0.0625

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� � ¡ ¢ £ ¤ ¥


GRAPHING LOGARITHMIC FUNCTIONS

By the definition of a logarithm, it follows that the logarithmic function g(x) = logb x

is the inverse of the exponential function ƒ(x) = bx. This means that:

g(ƒ(x)) = logb bx = x and ƒ(g(x)) = blogbx = x

In other words, exponential functions and logarithmic functions “undo” each other.

Using Inverse Properties


a. 10log 2 b. log3 9x

SOLUTION

a. 10log 2 = 2 b. log3 9x = log3 (32)x = log3 32x = 2x

Finding Inverses

Find the inverse of the function.

a. y = log3 x b. y = ln (x + 1)

SOLUTION

a. From the definition of logarithm, the inverse of y = log3 x is y = 3x.

b. y = ln (x + 1) Write original function.

x = ln (y + 1) Switch x and y.

ex = y + 1 Write in exponential form.

ex º 1 = y Solve for y.

c The inverse of y = ln (x + 1) is y = ex º 1.

. . . . . . . . . .

The inverse relationship between exponential and logarithmic functions is also usefulfor graphing logarithmic functions. Recall from Lesson 7.4 that the graph of ƒº1 isthe reflection of the graph of f in the line y = x.

Graphs of ƒ and ƒº1 for b > 1 Graphs of ƒ and ƒº1 for 0 < b < 1

¦§f (x ) 5 b x

f 21(x ) 5 logb x

¦§f (x ) 5 b x

f 21(x ) 5 logb x

E X A M P L E 6

E X A M P L E 5

GOAL 2

Look Back For help with inverses,see p. 422.

STUDENT HELP

¨ © ª « ¬ ® ¯


Graphing Logarithmic Functions


a. y = log1/3 x º 1 b. y = log5 (x + 2)

SOLUTION

a. Plot several convenient points,

such as S}

13

}, 0D and (3, º2). The

vertical line x = 0 is an asymptote.From left to right, draw a curve that starts just to the right of the y-axis and moves down.

The domain is x > 0, and the rangeis all real numbers.

Using the Graph of a Logarithmic Function

SCIENCE CONNECTION Graph the model from Example 4,

s = 0.159 + 0.118 log d. Then use the graph to estimate the average

diameter of the sand particles for a beach whose slope is 0.2.

SOLUTION

You can use a graphing calculator to graph the model.Then, using the Intersect feature, you can determinethat s = 0.2 when d ≈ 2.23, as shown at the right. So,the average diameter of the sand particles is about2.23 millimeters.

E X A M P L E 8

19s , 1d°± ¦§

(1, 21)

(3, 22)

y 5 log1/3 x 2 1

13s , 0d

E X A M P L E 7

The graph of y = logb (x º h) + k has the following characteristics:

• The line x = h is a vertical asymptote.

• The domain is x > h, and the range is all real numbers.

• If b > 1, the graph moves up to the right. If 0 < b < 1, the graph moves down

to the right.

GRAPHS OF LOGARITHMIC FUNCTIONS

Intersection

X=2.2256539 Y=.2

SAND A beach withvery fine sand makes

an angle of about 1° with thehorizontal while a beachwith pebbles makes an angleof about 17°.

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b. Plot several convenient points, such as (º1, 0) and (3, 1). Thevertical line x = º2 is an asymptote.From left to right, draw a curve thatstarts just to the right of the linex = º2 and moves up.

The domain is x > º2, and therange is all real numbers.

±° ²³y 5 log5 (x 1 2)

(21, 0)(3, 1)

s2 , 21d95

CONCEPT

SUMMARY

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1. Complete this statement: The logarithm with base 10 is called the ooo

? .

2. Explain why the expressions log3 (º1) and log1 1 are not defined.

3. Explain the meaning of logb y.

4. ERROR ANALYSIS To simplify log2 25, a student reasoned as shown. Describe the error that the student made.

Rewrite the equation in exponential form.

5. log3 9 = 2 6. log5 5 = 1 7. log1/2 4 = º2 8. log19 1 = 0

Evaluate the expression.

9. log2 64 10. log25 5 11. log6 1 12. 10log 4


13. y = log2 (x + 1) º 3 14. y = log1/2 (x º 2) + 1

15. SLOPE OF A BEACH Using the model from Example 4 and a graphingcalculator, find the average diameter of the sand particles for a beach whose

slope is 0.1.

REWRITING IN EXPONENTIAL FORM Rewrite the equation in exponential form.

16. log4 1024 = 5 17. log5 }

15

} = º1 18. log36 }

16

} = º}

12

} 19. log8 512 = 3

20. log12 144 = 2 21. log14 196 = 2 22. log8 4096 = 4 23. log105 11,025 = 2

EVALUATING EXPRESSIONS Evaluate the expression without using a

calculator.

24. log5 125 25. log7 343 26. log8 1 27. log12 12

28. log6 36 29. log4 16 30. log9 729 31. log7 2401

32. log1/4 }

14

} 33. log4 4º0.38 34. log4 }

21

} 35. log1/5 25

EVALUATING LOGARITHMS Use a calculator to evaluate the expression.

Round the result to three decimal places.

36. log 8 37. ln 10 38. log Ï2w 39. log 3.724

40. log 2.54 41. log 0.3 42. log 4.05 43. log 3.5

44. ln 4.6 45. ln 150 46. ln 6.9 47. ln 22.5

USING INVERSES Simplify the expression.

48. 5log5 x 49. log2 2x 50. 9log9

x 51. 35log35x

52. log4 16x 53. 7log7x 54. log 100x 55. log20 8000x


GUIDED PRACTICE

Vocabulary Check ✓Concept Check ✓

Skill Check ✓

STUDENT HELP

Extra Practice to help you masterskills is on p. 951.

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 16–23Example 2: Exs. 24–35Example 3: Exs. 36–47Example 4: Exs. 77–79Example 5: Exs. 48–55Example 6: Exs. 56–64Example 7: Exs. 65–76Example 8: Exs. 80, 81

52 = 25so log2 25 = 5

´ µ ¶ · ¸ ¹ º »


FINDING INVERSES Find the inverse of the function.

56. y = log9 x 57. y = log1/4 x 58. y = log5 x

59. y = log1/2 x 60. y = log7 49x 61. y = ln 6x

62. y = ln (x º 1) 63. y = ln (x + 2) 64. y = ln (x º 2)


65. y = log5 x 66. y = ln x + 3 67. y = log2 x + 1

68. y = ln x º 1 69. y = log8 x º 2 70. y = ln (x + 1)

71. y = log (x º 2) 72. y = ln (x º 2) 73. y = log5 (x + 4)

74. y = log1/2 x º 1 75. y = log1/4 x º 3 76. y = ln x + 5

77. The pH of a solution is given by the formula

pH = ºlog [H+]

where [H+] is the solution’s hydrogen ion concentration (in moles per liter). Find the pH of the solution.

a. lemon juice: [H+] = 1 ª 10º2.4 moles per liter

b. vinegar: [H+] = 1 ª 10º3 moles per liter

c. orange juice: [H+] = 1 ª 10º3.5 moles per liter

78. Part of the three-dimensional mathematical figure called thehorn of Gabriel is shown. The area of thecross section (in the coordinate plane) of thehorn is given by:

A = }lo

2g e}

Approximate this area to three decimal places.

SEISMOLOGY In Exercises 79 and 80, use the following information.

The Richter scale is used for measuring the magnitude of an earthquake. The Richtermagnitude R is given by the model

R = 0.67 log (0.37E) + 1.46

where E is the energy (in kilowatt-hours) released by the earthquake.

79. Suppose an earthquake releases 15,500,000,000 kilowatt-hours of energy. Whatis the earthquake’s magnitude? (Use a calculator.)

80. How many kilowatt-hours of energy would the earthquake in Exercise 79 have torelease in order to increase its magnitude by one-half of a unit on the Richterscale? Use a graph to solve the problem.

81. TORNADOES Most tornadoes last less than an hour and travel less than 20 miles. The wind speed s (in miles per hour) near the center of a tornado isrelated to the distance d (in miles) the tornado travels by this model:

s = 93 log d + 65

On March 18, 1925, a tornado whose wind speed was about 280 miles per hourstruck the Midwest. Use a graph to estimate how far the tornado traveled.

GEOMETRY CONNECTION

SCIENCE CONNECTION

¼ ½ ¾ ¿ 1xy 5

Ày 5 2

1x

Á¾

TORNADOES

form along aboundary between warmhumid air from the Gulf ofMexico and cool dry air fromthe north. When thunder-storm clouds develop alongthis boundary, the result isviolent weather which canproduce a tornado.

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´ µ ¶ · ¸ ¹ º »QUANTITATIVE COMPARISON In Exercises 82–87 choose the statement that is

true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

82.

83.

84.

85.

86.

87.

EVALUATING EXPRESSIONS Evaluate the expression. (Hint: Each expression

has the form logb x. Rewrite the base b and the x-value as powers of the same

number.)

88. log16 8 89. log16 64 90. log9 27 91. log4 512

92. CRITICAL THINKING What pattern do you recognize in your answers toExercises 88–91?

EVALUATING NUMERICAL EXPRESSIONS Evaluate the numerical expression.

(Review 6.1 for 8.5)

93. 52 • 53 94. (3º4)2 95. 70 • 73 • 7º2 96. S}

37

}Dº2

97. 98. S}

38

}Dº399. (º23)2 100. S}

45

}D3

101. S}

12

}Dº4102. (º32)º1 103. }

2

2

5

9} 104. S}

79

}Dº2

USING LONG DIVISION Divide using long division. (Review 6.5)

105. (2x2 + x º 1) ÷ (x + 4) 106. (x2 º 5x + 4) ÷ (x º 1)

107. (4x3 + 3x2 + 2x º 3) ÷ (x2 + 2) 108. (6x3 º 8x2 + 7) ÷ (x º 3)

FINDING A CUBIC MODEL Write a cubic function whose graph passes

through the given points. (Review 6.9)

109. (2, 0), (º3, 0), (0, 0), (3, º3) 110. (3, 0), (2, 0), (º3, 0), (0, º1)

111. (4, 0), (6, 0), (º4, 0), (1, 1) 112. (º2, 0), (º3, 0), (3, 0), (0, 2)

63

}

64

MIXED REVIEW


TestPreparation

★★Challenge

Column A Column B

log9 92/3 log 100

log16 1 0

log4 16 log8 64

ƒ(8) if ƒ(x) = log2 x 4

ƒ(º1) if ƒ(x) = log5 5x º1

ƒS}

12

}D if ƒ(x) = log3 9x log3 81

EXTRA CHALLENGE


Â Ã Ä Å Æ Ç È É

8.5 Properties of Logarithms 493

Properties of Logarithms

USING PROPERTIES OF LOGARITHMS

Because of the relationship between logarithms and exponents, you might expect logarithmsto have properties similar to the properties of exponents you studied in Lesson 6.1.

In the activity you may have discovered one of the properties of logarithms listed below.

Using Properties of Logarithms

Use log5 3 ≈ 0.683 and log5 7 ≈ 1.209 to approximate the following.

a. log5 }

37

} b. log5 21 c. log5 49

SOLUTION

a. log5 }

37

} = log5 3 º log5 7 ≈ 0.683 º 1.209 = º0.526

b. log5 21 = log5 (3 • 7) = log5 3 + log5 7 ≈ 0.683 + 1.209 = 1.892

c. log5 49 = log5 72 = 2 log5 7 ≈ 2(1.209) = 2.418

E X A M P L E 1

GOAL 1

Use properties of

logarithms.

Use properties of

logarithms to solve real-life


energy needed for molecular

transport in Exs. 77–79.

. To model real-life

quantities, such as the

loudness of different sounds

in Example 5.


GOAL 2

GOAL 1


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Investigating a Property of Logarithms

Copy and complete the table one row at a time.

Use the completed table to write a conjecture about the relationship among

logb u, logb v, and logb uv.

2

1

DevelopingConcepts

ACTIVITY

Let b, u, and v be positive numbers such that b ≠ 1.

PRODUCT PROPERTY logb uv = logb u + logb v

QUOTIENT PROPERTY logb }

u

v} = logb u º logb v

POWER PROPERTY logb un = n logb u

PROPERTIES OF LOGARITHMS

logb u logb v logb uv

log 10 = ? log 100 = ? log 1000 = ?

log 0.1 = ? log 0.01 = ? log 0.001 = ?

log2 4 = ? log2 8 = ? log2 32 = ?

Airport workers wear

hearing protection

because of the

loudness of jet engines.

Ê Ë Ì Í Î Ï Ð Ñ


You can use the properties of logarithms to expand and condense logarithmicexpressions.

Expanding a Logarithmic Expression

Expand log2 }

7yx3

}. Assume x and y are positive.

SOLUTION

log2 }

7yx3

} = log2 7x3 º log2 y Quotient property

= log2 7 + log2 x3 º log2 y Product property

= log2 7 + 3 log2 x º log2 y Power property

Condensing a Logarithmic Expression

Condense log 6 + 2 log 2 º log 3.

SOLUTION

log 6 + 2 log 2 º log 3 = log 6 + log 22 º log 3 Power property

= log (6 • 22) º log 3 Product property

= log }6 •

322

} Quotient property

= log 8 Simplify.

. . . . . . . . . .

Logarithms with any base other than 10 or e can be written in terms of common ornatural logarithms using the change-of-base formula.

Using the Change-of-Base Formula

Evaluate the expression log3 7 using common and natural logarithms.

SOLUTION

Using common logarithms: log3 7 = }l

l

o

o

g

g

7

3} ≈ }

00

.

.84

47

57

11

} ≈ 1.771

Using natural logarithms: log3 7 = }llnn

73

} ≈ }11

.

.90

49

69

} ≈ 1.771

E X A M P L E 4

E X A M P L E 3

E X A M P L E 2

Let u, b, and c be positive numbers with b ≠ 1 and c ≠ 1. Then:

logc u = }l

l

o

o

g

gb

b

u

c}

In particular, logc u = }l

l

o

o

g

g

u

c} and logc u = }

llnn

u

c}.

CHANGE-OF-BASE FORMULA

STUDENT HELP

Study TipWhen you are expandingor condensing anexpression involvinglogarithms, you mayassume the variables arepositive.

Ê Ë Ì Í Î Ï Ð Ñ


USING LOGARITHMIC PROPERTIES IN REAL LIFE

Using Properties of Logarithms

The loudness L of a sound (in decibels) isrelated to the intensity I of the sound (inwatts per square meter) by the equation

L = 10 log }

II

0}

where I0 is an intensity of 10º12 watt persquare meter, corresponding roughly to thefaintest sound that can be heard by humans.

a. Two roommates each play their stereos

at an intensity of 10º5 watt per squaremeter. How much louder is the musicwhen both stereos are playing, comparedwith when just one stereo is playing?

b. Generalize the result from part (a) byusing I for the intensity of each stereo.

SOLUTION

Let L1 be the loudness when one stereo is playing and let L2 be the loudness when both stereos are playing.

a. Increase in loudness = L2 º L1

= 10 log º 10 log }1

1

0

0

º

º

1

5

2} Substitute for L2 and L1.

= 10 log (2 • 107) º 10 log 107 Simplify.

= 10(log 2 + log 107 º log 107) Product property

= 10 log 2 Simplify.

≈ 3 Use a calculator.

c The sound is about 3 decibels louder.

b. Increase in loudness = L2 º L1

= 10 log }10

2

º

I

12} º 10 log }

10º

I

12}

= 10Slog }10

2º

I12

} º log }10º

I12

}D= 10Slog 2 + log }

10º

I12

} º log }10º

I12

}D= 10 log 2

≈ 3

c Again, the sound is about 3 decibels louder. This result tells you that theloudness increases by 3 decibels when both stereos are played regardless ofthe intensity of each stereo individually.

2 • 10º5

}

10º12

E X A M P L E 5

GOAL 2

SOUND

TECHNICIAN

Sound technicians operatetechnical equipment toamplify, enhance, record,mix, or reproduce sound.They may work in radio ortelevision recording studiosor at live performances.

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Decibel level Example

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Acoustics

130 Jet airplane takeoff

120 Riveting machine

110 Rock concert

100 Boiler shop

90 Subway train

80 Average factory

70 City traffic

60 Conversational speech

50 Average home

40 Quiet library

30 Soft whisper

20 Quiet room

10 Rustling leaf

0 Threshold of hearing

Ò Ó Ô Õ Ö × Ø Ù


1. Give an example of the property of logarithms.

a. product property b. quotient property c. power property

2. Which is equivalent to log S}

79

}D2? Explain.

A. 2(log 7 º log 9) B. }

2

lo

lo

g

g

9

7} C. Neither A nor B

3. Which is equivalent to log8 (5x2 + 3)? Explain.

A. log8 5x2 + log8 3 B. log8 5x2 • log8 3 C. Neither A nor B

4. Describe two ways to find the value of log6 11 using a calculator.

Use a property of logarithms to evaluate the expression.

5. log3 (3 • 9) 6. log2 45 7. log3 }

13

} 8. log5 S}

15

}D3

Use log2 7 ≈ 2.81 and log2 21 ≈ 4.39 to approximate the value of the expression.

9. log2 3 10. log2 49 11. log2 147 12. log2 441

13. SOUND INTENSITY Use the loudness of sound equation in Example 5 tofind the difference in the loudness of an average office with an intensity of1.26 ª 10º7 watt per square meter and a broadcast studio with an intensity of3.16 ª 10º10 watt per square meter.

EVALUATING EXPRESSIONS Use a property of logarithms to evaluate the

expression.

14. log2 (4 • 16) 15. ln eº2 16. log2 43 17. log5 125

18. log3 94 19. log }

110} 20. ln }

e

13} 21. log (0.01)3

APPROXIMATING EXPRESSIONS Use log 5 ≈ 0.699 and log 15 ≈ 1.176 to

approximate the value of the expression.

22. log 3 23. log 25 24. log 75 25. log 125

26. log }

15

} 27. log 225 28. log }

115} 29. log }

31

}

EXPANDING EXPRESSIONS Expand the expression.

30. log2 9x 31. ln 22x 32. log 4x5 33. log6 x6

34. log4 }

43

} 35. log3 25 36. log6 }

130} 37. ln 3xy3

38. log 6x3yz 39. log8 64x2 40. ln x1/2y3 41. log3 125/6x9

42. log Ïxw 43. ln 44. log Ï4x3w 45. log2 Ï4wxw3y4

}

x3


GUIDED PRACTICE


Concept Check ✓

Skill Check ✓


STUDENT HELP

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 14–29Example 2: Exs. 30–45Example 3: Exs. 46–57Example 4: Exs. 58–73Example 5: Exs. 74–85

Ò Ó Ô Õ Ö × Ø ÙCONDENSING EXPRESSIONS Condense the expression.

46. log5 8 º log5 12 47. ln 16 º ln 4

48. 2 log x + log 5 49. 4 log16 12 º 4 log16 2

50. 3 ln x + 5 ln y 51. 7 log4 2 + 5 log4 x + 3 log4 y

52. ln 20 + 2 ln }

12

} + ln x 53. log3 2 + }

12

} log3 y

54. 10 log x + 2 log 10 55. 3(ln 3 º ln x) + (ln x º ln 9)

56. 2(log6 15 º log6 5) + }

12

} log6 }

215} 57. }

14

} log5 81 º S2 log5 6 º }

12

} log5 4DCHANGE-OF-BASE FORMULA Use the change-of-base formula to evaluate

the expression.

58. log5 7 59. log7 12 60. log3 16 61. log9 25

62. log2 5 63. log6 9 64. log3 17 65. log5 32

66. log2 125 67. log6 24 68. log4 19 69. log16 81

70. log8 }

272} 71. log9 }

156} 72. log2 }

145} 73. log5 }

332}

PHOTOGRAPHY In Exercises 74–76, use the following information.

The f-stops on a 35 millimeter camera control the amount of light that enters thecamera. Let s be a measure of the amount of light that strikes the film and let ƒbe the f-stop. Then s and ƒ are related by this equation:

s = log2 ƒ2

74. Expand the expression for s.

75. The table shows the first eight f-stops on a 35 millimeter camera. Copy andcomplete the table. Then describe the pattern.

76. Many 35 millimeter cameras have nine f-stops. What do you think the ninth f-stop is? Explain your reasoning.

In Exercises 77–79, use the following information.

The energy E (in kilocalories per gram-molecule) required to transport a substancefrom the outside to the inside of a living cell is given by

E = 1.4(log C2 º log C1)

where C2 is the concentration of the substance inside the cell and C1 is theconcentration outside the cell.

77. Condense the expression for E.

78. The concentration of a particular substance inside a cell is twice theconcentration outside the cell. How much energy is required to transport thesubstance from outside to inside the cell?

79. The concentration of a particular substance inside a cell is six times theconcentration outside the cell. How much energy is required to transport the substance from outside to inside the cell?

SCIENCE CONNECTION


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ƒ 1.414 2.000 2.828 4.000 5.657 8.000 11.314 16.000

s ? ? ? ? ? ? ? ?

PHOTOGRAPHY

Photographers use f-stops to achieve thedesired amount of light in aphoto. The smaller the f-stopnumber, the more light thelens transmits.

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HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor help with Exs. 77–79.

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ACOUSTICS In Exercises 80–85, use the table and the loudness of sound

equation from Example 5.

80. The intensity of the sound made by a propeller aircraft is 0.316 watts per squaremeter. Find the decibel level of a propeller aircraft. To what sound in the tablefrom Example 5 is a propeller aircraft’s sound most similar?

81. The intensity of the sound made by Niagara Falls is 0.003 watts per square meter.Find the decibel level of Niagara Falls. To what sound in the table fromExample 5 is the sound of Niagara Falls most similar?

82. Three groups of people are in a room, and each group is having a conversation atan intensity of 1.4 ª 10º7 watt per square meter. What is the decibel level of thecombined conversations in the room?

83. Five cars are in a parking garage, and the sound made by each running car is atan intensity of 3.16 ª 10º4 watt per square meter. What is the decibel level ofthe sound produced by all five cars in the parking garage?

84. A certain sound has an intensity of I watts per square meter. By how manydecibels does the sound increase when the intensity is tripled?

85. A certain sound has an intensity of I watts per square meter. By how manydecibels does the sound decrease when the intensity is halved?

86. CRITICAL THINKING Tell whether this statement is true or false:log (u + v) = log u + log v. If true, prove it. If false, give a counterexample.

87. Writing Let n be an integer from 1 to 20. Use only the fact that log 2 ≈ 0.3010and log 3 ≈ 0.4771 to find as many values of log n as you possibly can. Showhow you obtained each value. What can you conclude about the values of n forwhich you cannot find log n?

88. MULTIPLE CHOICE Which of the following is not correct?

¡A log2 24 = log2 6 + log2 4 ¡B log2 24 = log2 72 º log2 3

¡C log2 24 = log2 8 + log2 16 ¡D log2 24 = 2 log2 2 + log2 6

89. MULTIPLE CHOICE Which of the following is equivalent to log5 8?

¡A }

l

l

o

o

g

g

5

8} ¡B }

l

l

o

o

g

g

8

5} ¡C }

llnn

85

} ¡D }

llnn

153

} ¡E Both B and C

90. MULTIPLE CHOICE Which of the following is equivalent to 4 log3 5?

¡A log3 20 ¡B log3 625 ¡C log3 60 ¡D log3 243 ¡E Both B and C

91. LOGICAL REASONING Use the given hint and properties of exponents to proveeach property of logarithms.

a. Product property (Hint: Let x = logb u and let y = logb v. Then u = bx andv = by so that logb uv = logb (bx • by ).)

b. Quotient property (Hint: Let x = logb u and let y = logb v. Then u = bx

and v = by so that logb = logb .)

c. Power property (Hint: Let x = logb u. Then u = bx and un = bnx so thatlogb un = logb (b

nx).)

d. Change-of-base formula (Hint: Let x = logb u, y = logb c, and z = logc u.Then u = bx, c = by, and u = c z so that bx = c z.)

bx

}

by

u}v

TestPreparation

★★Challenge

EXTRA CHALLENGE


FOCUS ON

APPLICATIONS

RALPH E. ALLISON

developed the firstsingle zero-point audiometerin 1937, making the equip-ment usable for doctors whohad previously used tuningforks to test hearing.

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SIMPLIFYING EXPRESSIONS Simplify the expression. (Review 6.1)

92. 3y2 • y2 93. (y4)3 94. (x3y)4 95. (º3x2)2

96. 4xº1y 97. xyº2x 98. }

x

xº

3

1} 99. }

8

4

x

x

y

2

º

y7

1}

SOLVING RADICAL EQUATIONS Solve the equation. Check for extraneous

solutions. (Review 7.6 for 8.6)

100. Ï4xw+w 2w + 9 = 14 101. Ï3

3wxwºw 4w = Ï3xw+w 1w0w

102. Ï3wxw+w 7w = x + 3 103. (5x)1/2 º 18 = 32

EVALUATING EXPRESSIONS Use a calculator to evaluate the expression.

Round the result to three decimal places. (Review 8.3, 8.4 for 8.6)

104. e9 105. eº12 106. e1.7 107. eº5.632

108. log 15 109. log 1.729 110. ln 16 111. ln 5.89

MIXED REVIEW

Logarithms

THENTHEN

William Oughtred

creates slide rule.

Modern day

calculator

IN 1614, John Napier published his discovery oflogarithms. This discovery allowed calculationswith exponents to be performed more easily. In1632 William Oughtred set two logarithmicscales side by side to form the first slide rule.Because the slide rule could be used to multiply,divide, raise to powers, and take roots, iteliminated the need for many tedious paper-and-pencil calculations.

1. To approximate the logarithm of a number,look at the number on the D row and the corresponding value on the L row of the slide rule shown above. For example, log 4 ≈ 0.6. Approximate log 3 and log 5.

2. Use the product property of logarithms to find log 15.

TODAY, calculators have replaced the use of slide rules but not the use of logarithms.Logarithms are still used for scaling purposes, such as the decibel scale and theRichter scale, because the numbers involved span many orders of magnitude.

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NOWNOW

16321999

John Napier invents

Napier’s Bones.

1617

D

L

Amendee Manheim

creates modern

slide rule.

1859

â ã ä å æ ç è éGraphingLogarithmic Functions

You can use a graphing calculator to graph logarithmic functions simply by

using the or key. To graph a logarithmic function having a base

other than 10 or e, you need to use the change-of-base formula to rewrite

the function in terms of common or natural logarithms.

c EXAMPLE

Use a graphing calculator to graph y = log2 x and y = log2 (x º 3) + 1.

c SOLUTION

Rewrite each function in terms of common logarithms.

y = log2 x y = log2 (x º 3) + 1

= }lloogg

2x

} = }

loglo(xg

º2

3)} + 1

Enter each function into a graphing calculator.

Graph the functions.

c EXERCISES

Use a graphing calculator to graph the function. Give the coordinates of a point

through which the graph passes, and state the vertical asymptote of the graph.

1. y = log3 x 2. y = log9 x 3. y = log4 x

4. y = log7 x 5. y = log5 x 6. y = log11 x

7. y = log5 (x º 2) 8. y = log4 (x + 1) 9. y = log2 (x º 5) º 3

10. y = log4 (x º 7) + 9 11. y = log5 (x + 2) + 6 12. y = log7 (x º 4) + 4

13. Compare the domains of the graphs of y = log x and y = log |x|.

3

Y1=(log X)/(log 2)Y2=(log (X-3))/((log 2)+1)Y3=Y4=Y5=Y6=

2

1


Using Technology

Graphing Calculator Activity for use with Lesson 8.5ACTIVITY 8.5

STUDENT HELP

KEYSTROKE

HELP

See keystrokes for several models ofcalculators atwww.mcdougallittell.com

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The graph of y = log2 x passesthrough (1, 0), and the line x = 0 is a vertical asymptote.

The graph of y = log2 (x – 3) + 1passes through (4, 1), and the line x = 3 is a vertical asymptote.

Although the calculator will correctlyevaluate the function without parentheses,you can include them for clarity.

ê ë ì í î ï ð ñ

8.6 Solving Exponential and Logarithmic Equations 501

Solve exponential

equations.

Solve logarithmic

equations, as applied in

Example 8.


problems, such as finding

the diameter of a telescope’s

objective lens or mirror

in Ex. 69.


GOAL 2

GOAL 1


8.6R

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Solving Exponential andLogarithmic Equations

SOLVING EXPONENTIAL EQUATIONS

One way to solve exponential equations is to use the property that if two powers withthe same base are equal, then their exponents must be equal.

For b > 0 and b ≠ 1, if bx = by, then x = y.

Solving by Equating Exponents

Solve 43x = 8x + 1.

SOLUTION

43x = 8x + 1 Write original equation.

(22)3x = (23)x + 1 Rewrite each power with base 2.

26x = 23x + 3 Power of a power property

6x = 3x + 3 Equate exponents.

x = 1 Solve for x.

c The solution is 1.

✓CHECK Check the solution by substituting it into the original equation.

43 • 1 · 81 + 1 Substitute 1 for x.

64 = 64 ✓ Solution checks.

. . . . . . . . . .

When it is not convenient to write each side of an exponential equation using thesame base, you can solve the equation by taking a logarithm of each side.

Taking a Logarithm of Each Side

Solve 2x = 7.

SOLUTION

2x = 7 Write original equation.

log2 2x = log2 7 Take log2 of each side.

x = log2 7 logb b x = x

x = }

l

l

o

o

g

g

7

2} ≈ 2.807 Use change-of-base formula and a calculator.

c The solution is about 2.807. Check this in the original equation.

E X A M P L E 2

E X A M P L E 1

GOAL 1

ê ë ì í î ï ð ñ


Taking a Logarithm of Each Side

Solve 102x º 3 + 4 = 21.

SOLUTION

102x º 3 + 4 = 21 Write original equation.

102x º 3 = 17 Subtract 4 from each side.

log 102x º 3 = log 17 Take common log of each side.

2x º 3 = log 17 log 10x = x

2x = 3 + log 17 Add 3 to each side.

x = }

12

}(3 + log 17) Multiply each side by }12}.

x ≈ 2.115 Use a calculator.

c The solution is about 2.115.

✓CHECK Check the solution algebraically by substituting into the original equation. Or, check it graphically by graphing both sides of the equation and observing that the two graphs intersect at x ≈ 2.115.

. . . . . . . . . .

Newton’s law of cooling states that the temperature T of a cooling substance at time t (in minutes) can be modeled by the equation

T = (T0 º TR)eºrt + TR

where T0 is the initial temperature of the substance, TR is the room temperature, and r is a constant that represents the cooling rate of the substance.

Using an Exponential Model

You are cooking aleecha, an Ethiopian stew. When you take it off the stove, itstemperature is 212°F. The room temperature is 70°F and the cooling rate of the stew isr = 0.046. How long will it take to cool the stew to a serving temperature of 100°F?

SOLUTION

You can use Newton’s law of cooling with T = 100, T0 = 212, TR = 70, and r = 0.046.

T = (T0 º TR)eºrt + TR Newton’s law of cooling

100 = (212 º 70)eº0.046t + 70 Substitute for T, T0, TR, and r.

30 = 142eº0.046tSubtract 70 from each side.

0.211 ≈ eº0.046tDivide each side by 142.

ln 0.211 ≈ ln eº0.046tTake natural log of each side.

º1.556 ≈ º0.046t ln e x = loge e x = x

33.8 ≈ t Divide each side by º0.046.

c You should wait about 34 minutes before serving the stew.

E X A M P L E 4

E X A M P L E 3

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Cooking

HOMEWORK HELP


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STUDENT HELP

ò ó ô õ ö ÷ ø ù


SOLVING LOGARITHMIC EQUATIONS

To solve a logarithmic equation, use this property for logarithms with the same base:

For positive numbers b, x, and y where b ≠ 1, logb x = logb y if and only if x = y.

Solving a Logarithmic Equation

Solve log3 (5x º 1) = log3 (x + 7).

SOLUTION

log3 (5x º 1) = log3 (x + 7) Write original equation.

5x º 1 = x + 7 Use property stated above.

5x = x + 8 Add 1 to each side.

x = 2 Solve for x.



log3 (5x º 1) = log3 (x + 7) Write original equation.

log3 (5 • 2 º 1) · log3 (2 + 7) Substitute 2 for x.

log3 9 = log3 9 ✓ Solution checks.

. . . . . . . . . .

When it is not convenient to write both sides of an equation as logarithmicexpressions with the same base, you can exponentiate each side of the equation.

For b > 0 and b ≠ 1, if x = y, then bx = by.

Exponentiating Each Side

Solve log5 (3x + 1) = 2.

SOLUTION

log5 (3x + 1) = 2 Write original equation.

5log5 (3x + 1) = 52 Exponentiate each side using base 5.

3x + 1 = 25 blogb x = x

x = 8 Solve for x.



log5 (3x + 1) = 2 Write original equation.

log5 (3 • 8 + 1) · 2 Substitute 8 for x.

log5 25 · 2 Simplify.

2 = 2 ✓ Solution checks.

E X A M P L E 6

E X A M P L E 5

GOAL 2

ò ó ô õ ö ÷ ø ù


Because the domain of a logarithmic function generally does not include all realnumbers, you should be sure to check for extraneous solutions of logarithmicequations. You can do this algebraically or graphically.

Checking for Extraneous Solutions

Solve log 5x + log (x º 1) = 2. Check for extraneous solutions.

SOLUTION

log 5x + log (x º 1) = 2 Write original equation.

log [5x(x º 1)] = 2 Product property of logarithms

10log (5x2 º 5x) = 102 Exponentiate each side using base 10.

5x2 º 5x = 100 10log x = x

x2 º x º 20 = 0 Write in standard form.

(x º 5)(x + 4) = 0 Factor.

x = 5 or x = º4 Zero product property

The solutions appear to be 5 and º4.However, when you check these in theoriginal equation or use a graphiccheck as shown at the right, you cansee that x = 5 is the only solution.


Using a Logarithmic Model

SEISMOLOGY The moment magnitude M of an earthquake that releases energy E (in ergs) can be modeled by this equation:

M = 0.291 ln E + 1.17

On May 22, 1960, a powerful earthquake took place in Chile. It had a momentmagnitude of 9.5. How much energy did this earthquake release?

c Source: U.S. Geological Survey National Earthquake Information Center

SOLUTION

M = 0.291 ln E + 1.17 Write model for moment magnitude.

9.5 = 0.291 ln E + 1.17 Substitute 9.5 for M.

8.33 = 0.291 ln E Subtract 1.17 from each side.

28.625 ≈ ln E Divide each side by 0.291.

e28.625 ≈ eln E Exponentiate each side using base e.

2.702 ª 1012 ≈ E eln x = eloge x = x

c The earthquake released about 2.7 trillion ergs of energy.

E X A M P L E 8

E X A M P L E 7

FOCUS ON

PEOPLE

CHARLES

RICHTER

developed the Richter scalein 1935 as a mathematicalmeans of comparing thesizes of earthquakes. Forlarge earthquakes, seismol-ogists use a different meas-ure called momentmagnitude.

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Look Back For help with the zeroproduct property, see p. 257.

STUDENT HELP

ú û ü ý þ ÿ � �1. Give an example of an exponential equation and a logarithmic equation.

2. How is solving a logarithmic equation similar to solving an exponentialequation? How is it different?

3. Why do logarithmic equations sometimes have extraneous solutions?

Solve the equation.

4. 3x = 14 5. 5x = 8 6. 92x = 3x º 6

7. 103x º 4 = 0.1 8. 23x = 4x º 1 9. 103x º 1 + 4 = 32

Solve the equation.

10. log x = 2.4 11. log x = 3 12. log3 (2x º 1) = 3

13. 12 ln x = 44 14. log2 (x + 2) = log2 x2 15. log 3x + log(x + 2)=1

ERROR ANALYSIS In Exercises 16 and 17, describe the error.

16. 4x + 1 = 8x 17. log2 5x = 8

log4 4x + 1 = log4 8

xelog2 5x = e8

x + 1 = x log4 8 5x = e8

x + 1 = 2x x = }

51} e

8

1 = x

18. EARTHQUAKES An earthquake that took place in Alaska on March 28,1964, had a moment magnitude of 9.2. Use the equation given in Example 8 to determine how much energy this earthquake released.

CHECKING SOLUTIONS Tell whether the x-value is a solution of the equation.

19. ln x = 27, x = e27 20. 5 º log4 2x = 3, x = 8

21. ln 5x = 4, x = }

14

}e5 22. log5 }

12

}x = 17, x = 2e17

23. 5ex = 15, x = ln 3 24. ex + 2 = 18, x = log2 16

SOLVING EXPONENTIAL EQUATIONS Solve the equation.

25. 10x º 3 = 1004x º 5 26. 25x º 1 = 1254x 27. 3x º 7 = 272x

28. 36x º 9 = 62x 29. 85x = 163x + 4 30. eºx = 6

31. 2x = 15 32. 1.2eº5x + 2.6 = 3 33. 4x º 5 = 3

34. º5eºx + 9 = 6 35. 102x + 3 = 8 36. 0.25x º 0.5 = 2

37. }

14

}(4)2x + 1 = 5 38. }

23

}e4x + }

13

} = 4 39. 10º12x + 6 = 100

40. 4 º 2ex = º23 41. 30.1x º 4 = 5 42. º16 + 0.2(10)x = 35


GUIDED PRACTICE


Vocabulary Check ✓Concept Check ✓

Skill Check ✓

STUDENT HELP

HOMEWORK HELP

Examples 1–3:Exs. 23–42

Example 4: Exs. 62–68Examples 5–7:

Exs. 19–22, 43–60Example 8: Exs. 69, 70


STUDENT HELP

ú û ü ý þ ÿ � �


SOLVING LOGARITHMIC EQUATIONS Solve the equation. Check for

extraneous solutions.

43. ln (4x + 1) = ln (2x + 5) 44. log2 x = º1

45. 4 log3 x = 28 46. 16 ln x = 30

47. }

12

} log6 16x = 3 48. 1 º 2 ln x = º4

49. 2 ln (ºx) + 7 = 14 50. log5 (2x + 15) = log5 3x

51. ln x + ln (x º 2) = 1 52. ln x + ln (x + 3) = 1

53. log8 (11 º 6x) = log8 (1 º x) 54. 15 + 2 log2 x = 31

55. º5 + 2 ln 3x = 5 56. log (5 º 3x) = log (4x º 9)

57. 6.5 log5 3x = 20 58. ln (x + 5) = ln (x º 1) º ln (x + 1)

59. ln (5.6 º x) = ln (18.4 º 2.6x) 60. 10 ln 100x º 3 = 117

61. Writing Solve the equation 43x = 8x + 1 in Example 1 by taking the commonlogarithm of each side of the equation. Do you prefer this method to the methodshown in Example 1? Why or why not?

62. COOKING You are cooking chili. When you take it off the stove, it has atemperature of 205°F. The room temperature is 68°F and the cooling rate of thechili is r = 0.03. How long will it take to cool to a serving temperature of 95°F?

63. FINANCE You deposit $2000 in an account that pays 2% annual interestcompounded quarterly. How long will it take for the balance to reach $2400?

64. RADIOACTIVE DECAY You have 20 grams of phosphorus-32 that decays 5% per day. How long will it take for half of the original amount to decay?

65. DOUBLING TIME You deposit $500 in an account that pays 2.5% annualinterest compounded continuously. How long will it take for the balance todouble?

66. The first permanent English colony in America wasestablished in Jamestown, Virginia, in 1607. From 1620 through 1780, thepopulation P of colonial America can be modeled by the equation

P = 8863(1.04)t

where t is the number of years since 1620. When was the population of colonialAmerica about 345,000?

67. OCEANOGRAPHY Oceanographersuse the density d (in grams per cubiccentimeter) of seawater to obtain informa-tion about the circulation of water massesand the rates at which waters of differentdensities mix. For water with a salinity of30%, the density is related to the watertemperature T (in degrees Celsius) by thisequation:

d = 1.0245 º e0.1266T º 7.828

Use the equation to find the temperature of each layer of water whose density isgiven in the diagram.

HISTORY CONNECTION

Subantarctic

Antarctic intermediate

North Atlanticdeep

Antarctic bottom

d 51.02340 g/cm3

d 5 1.02384 g/cm3

d 5 1.02399 g/cm3

d 5 1.02410 g/cm3

Antarcticsurface

Antarctic

convergence

� � � � � � �


68. MUON DECAY A muon is an elementary particle that is similar to anelectron, but much heavier. Muons are unstable—they very quickly decay toform electrons and other particles. In an experiment conducted in 1943, thenumber m of muon decays (of an original 5000 muons) was related to the time t(in microseconds) by this model:

m = e6.331 º 0.403t

After how many microseconds were 204 decays recorded?

69. ASTRONOMY The relationship between a telescope’s limiting magnitude(the apparent magnitude of the dimmest star that can be seen with the telescope)and the diameter of the telescope’s objective lens or mirror can be modeled by

M = 5 log D + 2

where M is the limiting magnitude and D is the diameter (in millimeters) of thelens or mirror. If a telescope can reveal stars with a magnitude of 12, what is thediameter of its objective lens or mirror? c Source: Practical Astronomy

70. ALTIMETER An altimeter is an instrument that finds the height above sealevel by measuring the air pressure. The height and the air pressure are related by the model

h = º8005 ln }101

P,300}

where h is the height (in meters) above sea level and P is the air pressure (inpascals). What is the air pressure when the height is 4000 meters above sea level?

71. MULTI-STEP PROBLEM A simpletechnique that biologists use to estimate theage of an African elephant is to measure thelength of the elephant’s footprint and thencalculate its age using the equation

l = 45 º 25.7eº0.09a

where l is the length of the footprint (incentimeters) and a is the age (in years).

c Source: Journal of Wildlife Management

a. Use the equation to find the ages of theelephants whose footprints are shown.

b. Solve the equation for a, and use thisequation to find the ages of the elephantswhose footprints are shown.

c. Writing Compare the methods youused in parts (a) and (b). Which methoddo you prefer? Explain.

SOLVING EQUATIONS Solve the equation.

72. 2x + 3 = 53x º 1 73. 105x + 2 = 54 º x

74. log3 (x º 6) = log9 2x 75. log4 x = log8 4x

76. Writing In Exercises 72–75 you solved exponential and logarithmic equationswith different bases. Describe general methods for solving such equations.

★★Challenge

TestPreparation

APPARENT

MAGNITUDE of astar is a number indicatingthe brightness of the star asseen from Earth. The greaterthe apparent magnitude, thefainter the star.

APPLICATION LINK


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FOCUS ON

APPLICATIONS

EXTRA CHALLENGE


36 cm

32 cm

28 cm

24 cm

� � � � � � � MAKING SCATTER PLOTS Draw a scatter plot of the data. Then approximate

an equation of the best-fitting line. (Review 2.5 for 8.7)

77.

78.

THE SUBSTITUTION METHOD Solve the linear system using the substitution

method. (Review 3.2 for 8.7)

79. 2x º y = 3 80. 2x + y = 4 81. x + 4y = º243x º 2y = 2 x + y = 3 x º 4y = 24

82. x º 3y = º3 83. 2x + y = º1 84. ºx + 6y = º322x + y = 8 º4x º 2y = º5 7x º 2y = 24

FACTORING Factor the polynomial by grouping. (Review 6.4)

85. 3x3 º 6x2 + 4x º 8 86. 2x3 º 5x2 + 16x º 40

87. 7x3 + 4x2 + 35x + 20 88. 4x3 º 3x2 + 8x º 6

Evaluate the expression without using a calculator. (Lesson 8.4)

1. log2 8 2. log5 625 3. log8 512

4. Find the inverse of the function y = ln (x + 3). (Lesson 8.4)

Graph the function. State the domain and range. (Lesson 8.4)

5. y = 1 + log4 x 6. y = log4 (x + 3) 7. y = 2 + log6 (x º 2)

Use a property of logarithms to evaluate the expression. (Lesson 8.5)

8. log3 (3 • 27) 9. log2 }

12

} 10. ln e2

11. Expand the expression log4 x1/2y4. (Lesson 8.5)

12. Condense the expression 2 log6 14 + 3 log6 x º log6 7. (Lesson 8.5)

13. Use the change-of-base formula to evaluate the expression log4 22. (Lesson 8.5)

Solve the equation. (Lesson 8.6)

14. 3e x º 1 = 14 15. 3 log2 x = 28 16. ln (2x + 7) = ln (x º 4)

17. EARTHQUAKES An earthquake that took place in Indonesia on February 1,1938, had a moment magnitude of 8.5. Use the model M = 0.291 ln E + 1.17,where M is the moment magnitude and E is the energy (in ergs) of an earthquake,to determine how much energy the Indonesian earthquake released. (Lesson 8.6)

QUIZ 2 Self-Test for Lessons 8.4–8.6

MIXED REVIEW


x º4 º3 º2.5 º2 º1.5 º1 0 1 1.5 2

y 1.5 1.75 1.75 2.25 2 2.25 2.75 2.75 3 3.5

x º2 º1 º0.5 0 0.5 1 2 3 3.5 4

y 1.25 1.5 1.5 2 1.75 2 2.5 2.5 2.75 3.25

� � � � � �

8.7 Modeling with Exponential and Power Functions 509

Modeling with Exponential and Power Functions

MODELING WITH EXPONENTIAL FUNCTIONS

Just as two points determine a line, two points also determine an exponential curve.

Writing an Exponential Function

Write an exponential function y = abx whose graph passes through (1, 6) and (3, 24).

SOLUTION

Substitute the coordinates of the two given points into y = abx to obtain twoequations in a and b.

6 = ab1 Substitute 6 for y and 1 for x.

24 = ab3 Substitute 24 for y and 3 for x.

To solve the system, solve for a in the first equation to get a = }

6b

}, then substitute intothe second equation.

24 = S}

6

b}Db3 Substitute }

b6} for a.

24 = 6b2 Simplify.

4 = b2 Divide each side by 6.

2 = b Take the positive square root.

Using b = 2, you then have a = }

6b

} = }

62

} = 3. So, y = 3 • 2x.

. . . . . . . . . .

When you are given more than two points, you can decide whether an exponentialmodel fits the points by plotting the natural logarithms of the y-values against the x-values. If the new points (x, ln y) fit a linear pattern, then the original points (x, y)fit an exponential pattern.

Graph of points (x, y) Graph of points (x, ln y)

The graph is an exponential curve. The graph is a line.

(1, 0.69)

(22, 21.39)

�y 5 x (ln 2)

(21, 20.69)(0, 0)

� ��

�� y 5 2x ��

s21, d12

(1, 2)

(0, 1)s22, d14

E X A M P L E 1

GOAL 1

Model data with

exponential functions.

Model data with

power functions, as applied

in Example 5.

. To solve real-lifeproblems, such as finding

the number of U.S. stamps

issued in Ex. 56.


GOAL 2

GOAL 1


8.7

E X P L O R I N G D ATA

A N D S TAT I S T I C S

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� � � � � � � �


Finding an Exponential Model

The table gives the number y (in millions) of cell-phone subscribers from 1988 to1997 where t is the number of years since 1987.

c Source: Cellular Telecommunications Industry Association

a. Draw a scatter plot of ln y versus x. Is an exponential model a good fit for theoriginal data?

b. Find an exponential model for the original data.

SOLUTION

a. Use a calculator to create a new table of values.

Then plot the new points as shown. The pointslie close to a line, so an exponential modelshould be a good fit for the original data.

b. To find an exponential model y = abt, choosetwo points on the line, such as (2, 0.99) and(9, 3.64). Use these points to find an equationof the line. Then solve for y.

ln y = 0.379t + 0.233 Equation of line

y = e0.379t + 0.233 Exponentiate each side using base e.

y = e0.233(e0.379)tUse properties of exponents.

y = 1.30(1.46)tExponential model

. . . . . . . . . .

A graphing calculator that performs exponential regression does essentially what isdone in Example 2, but uses all of the original data.

Using Exponential Regression

Use a graphing calculator to find an exponential model for the data in Example 2.

Use the model to estimate the number of cell-phone subscribers in 1998.

SOLUTION

Enter the original data into a graphing calculator andperform an exponential regression. The model is:

y = 1.30(1.46)t

Substituting t = 11 (for 1998) into the model gives y = 1.30(1.46)11 ≈ 84 million cell-phone subscribers.

E X A M P L E 3

E X A M P L E 2

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CommunicationsR

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Communications

t 1 2 3 4 5 6 7 8 9 10

y 1.6 2.7 4.4 6.4 8.9 13.1 19.3 28.2 38.2 48.7

t 1 2 3 4 5 6 7 8 9 10

In y 0.47 0.99 1.48 1.86 2.19 2.57 2.96 3.34 3.64 3.89

ExpReg y=a*bˆx a=1.30076406 b=1.458520596

r2=.9934944894 r=.9967419372

(9, 3.64)

(2, 0.99) !

" # $Look Back For help with scatterplots and best-fittinglines, see pp. 100–101.

STUDENT HELP

� � � � � � � �


MODELING WITH POWER FUNCTIONS

Recall from Lesson 7.3 that a power function has the form y = axb. Because thereare only two constants (a and b), only two points are needed to determine a powercurve through the points.

Writing a Power Function

Write a power function y = axb whose graph passes through (2, 5) and (6, 9).

SOLUTION

Substitute the coordinates of the two given points into y = axb to obtain twoequations in a and b.

5 = a • 2bSubstitute 5 for y and 2 for x.

9 = a • 6bSubstitute 9 for y and 6 for x.

To solve the system, solve for a in the first equation to get a = }

2

5b}, then substitute

into the second equation.

9 = S}

2

5

b}D6b

Substitute }2

5b} for a.

9 = 5 • 3bSimplify.

1.8 = 3bDivide each side by 5.

log3 1.8 = b Take log3 of each side.

}

lo

lo

g

g

1

3

.8} = b Use the change-of-base formula.

0.535 ≈ b Use a calculator.

Using b = 0.535, you then have a = }

2

5b} = }

20

5.535} ≈ 3.45. So, y = 3.45x0.535.

. . . . . . . . . .

When you are given more than two points, you can decide whether a powermodel fits the points by plotting the natural logarithms of the y-values againstthe natural logarithms of the x-values. If the new points (ln x, ln y) fit a linearpattern, then the original points (x, y) fit a power pattern.

Graph of points (x, y) Graph of points (ln x, ln y)

The graph is a power curve. The graph is a line.

ln y 5 ln x12

(1.10, 0.55)

%(0, 0)

(1.79, 0.9)

(1.39, 0.69)

" # $" # & (1, 1)

(4, 2)(3, 1.73)

(6, 2.45)

y 5 x 1/2 &" # $

E X A M P L E 4

GOAL 2

' ( ) * + , - .Finding a Power Model

The table gives the mean distance x from the sun (in astronomical units) and theperiod y (in Earth years) of the six planets closest to the sun.

a. Draw a scatter plot of ln y versus ln x. Is a power model a good fit for theoriginal data?

b. Find a power model for the original data.

SOLUTION

a. Use a calculator to create a new table of values.

Then plot the new points, as shown at the right. The points lie close to a line, so a power model should be a good fit for the original data.

b. To find a power model y = axb, choose two points on the line, such as (0, 0) and (2.255, 3.383). Use these points to find an equation of the line. Then solve for y.

ln y = 1.5 ln x Equation of line

ln y = ln x1.5 Power property of logarithms

y = x1.5 logb x = logb y if and only if x = y.

. . . . . . . . . .

A graphing calculator that performs power regression does essentially what is done inExample 5, but uses all of the original data.

Using Power Regression

ASTRONOMY Use a graphing calculator to find a power model for the data in

Example 5. Use the model to estimate the period of Neptune, which has a mean

distance from the sun of 30.043 astronomical units.

SOLUTION

Enter the original data into a graphing calculator and perform a power regression. The model is:

y = x1.5

Substituting 30.043 for x in the model givesy = (30.043)1.5 ≈ 165 years for the period of Neptune.

E X A M P L E 6

E X A M P L E 5


PwrReg y=a*xˆb a=1.000276492 b=1.499649516

r2=.9999999658 r=.9999999829

" # $ (0, 0) " # &(2.255, 3.383)

Planet Mercury Venus Earth Mars Jupiter Saturn

x 0.387 0.723 1.000 1.524 5.203 9.539

y 0.241 0.615 1.000 1.881 11.862 29.458

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Astronomy

In x º0.949 º0.324 0.000 0.421 1.649 2.255

In y º1.423 º0.486 0.000 0.632 2.473 3.383

JOHANNES

KEPLER, a Germanastronomer and mathemati-cian, was the first person toobserve that a planet’sdistance from the sun and its period were related by the power function inExamples 5 and 6.

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FOCUS ON

PEOPLE

' ( ) * + , - .1. Complete this statement: When you are given more than two points, you can

decide whether you can fit a(n) ooo

? model to the points by plotting the naturallogarithms of the y-values against the x-values.

2. How many points determine an exponential function y = abx? How many pointsdetermine a power function y = axb?

3. Can you use the procedure in Example 5 to find a power model for a data setwhere one of the points has an x-coordinate of 0? Explain why or why not.

Write an exponential function of the form y = abx whose graph passes through

the given points.

4. (1, 3), (2, 36) 5. (2, 2), (4, 18) 6. (1, 4), (3, 16)

7. (2, 3.5), (1, 5.2) 8. (5, 8), (3, 32) 9. S1, }

12

}D, S3, }

38

}DWrite a power function of the form y = axb whose graph passes through the

given points.

10. (3, 27), (9, 243) 11. (1, 2), (4, 32) 12. (4, 48), (2, 6)

13. (1, 4), (3, 8) 14. (4.5, 9.2), (1, 6.4) 15. S2, }

12

}D, S4, }

35

}D16. CELL-PHONE USERS Use the model in Example 3 to estimate the number

of cell-phone users in 2005. What does your answer tell you about the model?

WRITING EXPONENTIAL FUNCTIONS Write an exponential function of the

form y = abx whose graph passes through the given points.

17. (1, 4), (2, 12) 18. (2, 18), (3, 108) 19. (6, 8), (7, 32)

20. (1, 7), (3, 63) 21. (3, 8), (6, 64) 22. (º3, 3), (4, 6561)

23. S4, }18112

}D, Sº1, }

221}D 24. (3, 13.5), (5, 30.375) 25. S2, }

245}D, S4, }

624

5}D

FINDING EXPONENTIAL MODELS Use the table of values to draw a scatter

plot of ln y versus x. Then find an exponential model for the data.

26.

27.

28.


GUIDED PRACTICE

x 1 2 3 4 5 6 7 8

y 14 28 56 112 224 448 896 1792


Concept Check ✓

Skill Check ✓

x 1 2 3 4 5 6 7 8

y 10.2 30.5 43.4 61.2 89.7 120.6 210.4 302.5

x 2 4 6 8 10 12 14 16

y 12.8 20.48 32.77 52.43 83.89 134.22 214.75 343.6


STUDENT HELP

STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 17–25Example 2: Exs. 26–28Example 3: Exs. 54–56Example 4: Exs. 29–37Example 5: Exs. 38–40Example 6: Exs. 57, 58


/ 0 1 2 3 4 5 6WRITING POWER FUNCTIONS Write a power function of the form y = axb

whose graph passes through the given points.

29. (2, 1), (6, 5) 30. (6, 8), (12, 36) 31. (5, 12), (7, 25)

32. (3, 4), (6, 18) 33. (2, 10), (8, 25) 34. (6, 11), (24, 72)

35. (2.2, 10.4), (8.8, 20.3) 36. (2.9, 9.4), (7.3, 12.8) 37. (2.71, 6.42), (13.55, 29.79)

FINDING POWER MODELS Use the table of values to draw a scatter plot of

ln y versus ln x. Then find a power model for the data.

38.

39.

40.

WRITING EQUATIONS Write y as a function of x.

41. log y = 0.24x + 4.5 42. log y = 0.2 log x + 0.8

43. ln y = x + 4 44. log y = º0.12 + 0.88x

45. log y = º0.48 log x º 0.548 46. ln y = 2.3 ln x + 4.7

47. ln y = º2.38x + 0.98 48. log y = º1.48 + 3.751 log x

49. ln y = º1.5x + 2.5 50. 1.2 log y = 3.4 log x

51. }

12

} log y = }

56

} log x 52. 2}

18

} ln y = 4}

14

} ln x + }

83

}

53. VISUAL THINKING Find equations of the line, the exponential curve, and thepower curve that each pass through the points (1, 3) and (2, 12). Graph theequations in the same coordinate plane and then describe what happens when theequations are used as models to predict y-values for x-values greater than 2.

MODELING DATA In Exercises 54–58, you may wish to use a graphing

calculator to perform exponential regression or power regression.

54. NEW WEB SITE You have just created your own Web site. You are keepingtrack of the number of hits (the number of visits to the site). The table shows thenumber y of hits in each of the first 10 months where x is the month number.

a. Find an exponential model for the data.

b. According to your model, how many hits do you expect in the twelfth month?

c. According to your model, how many hits would there be in the thirty-fourthmonth? What is wrong with this number?


x 1 2 3 4 5 6 7 8 9 10

y 22 39 70 126 227 408 735 1322 2380 4285

x 1 2 3 4 5 6 7

y 0.78 7.37 27.41 69.63 143.47 259.00 426.79

x 1 2 3 4 5 6 7

y 1.2 5.4 9.8 14.3 25.6 41.2 65.8

x 2 4 6 8 10 12 14

y 1.89 1.44 1.22 1.09 1.00 0.93 0.87

/ 0 1 2 3 4 5 655. CRANES The table shows the number C of cranes in Izumi, Japan, from

1950 to 1990 where t represents the number of years since 1950.

c Source: Yamashina Institute of Ornithology

a. Draw a scatter plot of ln C versus t. Is an exponential model a good fit for theoriginal data?

b. Find an exponential model for the original data. Estimate the number ofcranes in Izumi, Japan, in the year 2000.

56. UNITED STATES STAMPS The table shows the cumulative number s ofdifferent stamps in the United States from 1889 to 1989 where t represents thenumber of years since 1889.

a. Draw a scatter plot of ln s versus t. Is an exponential model a good fit for theoriginal data?

b. Find an exponential model for the original data. Estimate the cumulativenumber of stamps in the United States in the year 2000.

57. CITIES OF ARGENTINA

The table shows the population y

(in millions) and the population rank x for nine cities in Argentina in 1991.

a. Draw a scatter plot of ln y

versus ln x. Is a power model agood fit for the original data?

b. Find a power model for the original data. Estimate thepopulation of the city VicenteLópez, which has a populationrank of 20.

58. The table shows the atomic number x and the meltingpoint y (in degrees Celsius) for the alkali metals.

a. Draw a scatter plot of ln y versus ln x. Is a power model a good fit for theoriginal data?

b. Find a power model for the original data.

c. One of the alkali metals, francium, is not shown in the table. It has an atomicnumber of 87. Using your model, predict the melting point of francium.

SCIENCE CONNECTION


t 0 5 10 15 20 25 30 35 40

C 293 299 438 1573 2336 3649 5602 7610 9959

City Rank, x Population(millions), y

Cordoba 2 1.21

Rosario 3 1.12

La Matanza 4 1.11

Mendoza 5 0.77

La Plata 6 0.64

Moron 7 0.64

San Miguel de 8 0.62Tucuman

Lomas de Zamoras 9 0.57

Mar de Plata 10 0.51

Alkali metal Lithium Sodium Potassium Rubidium Cesium

Atomic number, x 3 11 19 37 55

Melting point, y 180.5 97.8 63.7 38.9 28.5

CRANES

The red-crownedcrane (Grus japonensis) isthe second-rarest cranespecies, with a totalpopulation in the wild ofabout 1700–2000 birds.

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APPLICATIONS

HOMEWORK HELP

Visit our Web sitewww.mcdougallittell.comfor help with Ex. 57.

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t 0 10 20 30 40 50 60 70 80 90 100

s 218 293 374 541 681 858 986 1138 1138 1794 2438

7 8 9 : ; < = >


59. MULTI-STEP PROBLEM The femuris a large bone found in the leg orhind limb of an animal. Scientists usethe circumference of an animal’sfemur to estimate the animal’s weight.The table at the right shows the femurcircumference C (in millimeters) andthe weight W (in kilograms) of severalanimals.

a. Draw two scatter plots, one of ln Wversus C and another of ln W

versus ln C.

b. Writing Looking at your scatterplots, tell which type of model you think is a better fit for the original data. Explain your reasoning.

c. Using your answer from part (b), find a model for the original data.

d. The table at the right shows the femur circumference C (in millimeters) of four animals. Use the model you found in part (c) to estimate the weight of each animal.

60. DERIVING FORMULAS Using y = abx and y = axb, take the natural logarithmof both sides of each equation. What is the slope and y-intercept of the linerelating x and ln y for y = abx? of the line relating ln x and ln y for y = axb?

DESCRIBING END BEHAVIOR Describe the end behavior of the graph of the

polynomial function by completing the statements ƒ(x) ˘ooo

? as x ˘ º‡ and

ƒ(x) ˘ ooo

? as x ˘ +‡. (Review 6.2 for 8.8)

61. ƒ(x) = ºx3 + x2 º x + 4 62. ƒ(x) = x4 º 7x2 + 2

63. ƒ(x) = ºx4 + 3x º 3 64. ƒ(x) = 3x5 º x4 º x2 + 1

65. ƒ(x) = x6 º 2x º 1 66. ƒ(x) = º2x5 + 3x4 º 2x3 + x2 + 5

GRAPHING FUNCTIONS Graph the function. (Review 8.3 for 8.8)

67. y = 4eº0.75x 68. y = 10eº0.4x 69. y = 2ex º 3

70. y = e0.5x + 2 71. y = eº0.25x º 4 72. y = 3eº1.5x º 1

73. y = 2e0.25x + 1 74. y = ex + 1 º 5 75. y = 2.5eº0.6x + 2

CONDENSING EXPRESSIONS Condense the expression. (Review 8.5)

76. 5 log 2 º log 8 77. 2 log 9 º log 3

78. ln x + 5 ln 3 79. 2 ln x º ln 4

80. log2 8 + 3 log2 3 º log2 6 81. log7 12 + 3 log7 4 + log7 5

MIXED REVIEW

TestPreparation

★★Challenge

Animal C (mm)

Raccoon 28

Cougar 60.25

Bison 167.5

Hippopotamus 208

c Source: Zoological Society of London

Animal C (mm) W (kg)

Meadow mouse 5.5 0.047

Guinea pig 15 0.385

Otter 28 9.68

Cheetah 68.7 38

Warthog 72 90.5

Nyala 97 134.5

Grizzly bear 106.5 256

Kudu 135 301

Giraffe 173 710

7 8 9 : ; < = >Logistic Growth Functions

USING LOGISTIC GROWTH FUNCTIONS

In this lesson you will study a family of functions of the form

y =

where a, c, and r are all positive constants. Functions of this form are called

Evaluating a Logistic Growth Function

Evaluate ƒ(x) = for each value of x.

a. ƒ(º3) b. ƒ(0) c. ƒ(2) d. ƒ(4)

SOLUTION

a. ƒ(º3) = ≈ 0.0275 b. ƒ(0) = = }1

1

0

0

0} = 10

c. ƒ(2) = ≈ 85.8 d. ƒ(4) = ≈ 99.7

In this chapter you learned that an exponential growthfunction ƒ(x) increases without bound as x increases. On the other hand, the logistic growth

function y = has y = c as an upper bound.

Logistic growth functions are used to model real-lifequantities whose growth levels off because the rate ofgrowth changes—from an increasing growth rate to adecreasing growth rate.

c}}

1 + aeºrx

100}}

1 + 9eº2(4)

100}}

1 + 9eº2(2)

100}}

1 + 9eº2(0)

100}}

1 + 9eº2(º3)

100}}

1 + 9eº2x

E X A M P L E 1

logistic growth functions.

c}}

1 + aeºrx

GOAL 1

8.8 Logistic Growth Functions 517

Evaluate and graph

logistic growth functions.

Use logistic growth

functions to model real-lifequantities, such as a yeast

population in Exs. 50 and 51.

. To solve real-lifeproblems, such as modeling

the height of a sunflower

in Example 5.


GOAL 2

GOAL 1


8.8R

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Graphs of Logistic Growth Functions

Use a graphing calculator to graph the logistic growth function from

Example 1. Trace along the graph to determine the function’s end behavior.

Use a graphing calculator to graph each of the following. Then describe the

basic shape of the graph of a logistic growth function.

a. y = b. y = c. y = 5

}}

1 + 10eº2x10

}}

1 + 5eº2x1

}

1 + eºx

2

1

DevelopingConcepts

ACTIVITY

?@y 5 c

increasinggrowthrate

decreasinggrowthrate

point of maximum growth

A B C D E F G H

Graphing a Logistic Growth Function

Graph y = .

SOLUTION

Begin by sketching the upper horizontal asymptote, y = 6. Then plot the y-intercept at (0, 2) and the point

of maximum growth S , D ≈ (1.4, 3). Finally,

from left to right, draw a curve that starts just abovethe x-axis, curves up to the point of maximum growth,and then levels off as it approaches the upperhorizontal asymptote.

Solving a Logistic Growth Equation

Solve = 40.

SOLUTION

= 40 Write original equation.

50 = (1 + 10eº3x)(40) Multiply each side by 1 + 10e–3x.

50 = 40 + 400eº3xUse distributive property.

10 = 400eº3xSubtract 40 from each side.

0.025 = eº3xDivide each side by 400.

ln 0.025 = º3x Take natural log of each side.

º ln 0.025 = x Divide each side by º3.

1.23 ≈ x Use a calculator.

c The solution is about 1.23. Check this in the original equation.

1}

3

50}}

1 + 10eº3x

50}}

1 + 10eº3x

E X A M P L E 3

6}

2ln 2}

0.5

6}}

1 + 2eº0.5x

E X A M P L E 2


The graph of y = has the following characteristics:

• The horizontal lines y = 0 and y = c are asymptotes.

• The y-intercept is .

• The domain is all real numbers, and the range is 0 < y < c.

• The graph is increasing from left to right. To the left of its point of

maximum growth, S , D, the rate of increase is increasing. To

the right of its point of maximum growth, the rate of increase is decreasing.

c}

2ln a}

r

c}

1 + a

c}}

1 + aeºrx

GRAPHS OF LOGISTIC GROWTH FUNCTIONS

IJKK y 5 6

(0, 2)(1.4, 3)

6

1 1 2e20.5xy 5

HOMEWORK HELP


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L M N O P Q R S


USING LOGISTIC GROWTH MODELS IN REAL LIFE

Logistic growth functions are often more useful as models than exponential growthfunctions because they account for constraints placed on the growth. An example is abacteria culture allowed to grow under initially ideal conditions, followed by lessfavorable conditions that inhibit growth.

Using a Logistic Growth Model

A colony of the bacteria B. dendroides is growing in a petri dish. The colony’s area A(in square centimeters) can be modeled by

A =

where t is the elapsed time in days. Graph the function and describe what it tells youabout the growth of the bacteria colony.

SOLUTION

The graph of the model is shown. The initial area is

A = ≈ 0.37 cm2.

The colony grows more and more rapidly until

t = ≈ 2.5 days.

Then the rate of growth decreases. The colony’sarea is limited to A = 49.9 cm2, which mightpossibly be the area of the petri dish.

Writing a Logistic Growth Model

You planted a sunflower seedling and kept track of its height h (in centimeters)

over time t (in weeks). Find a model that gives h as a function of t.

SOLUTION

A scatter plot shows that the data can be modeled by a logistic growth function.

The logistic regression feature of a graphing calculator returns the values shown at the right.

c The model is:

h = 256

}}

1 + 13eº0.65t

E X A M P L E 5

ln 134}

1.96

49.9}}

1 + 134eº1.96(0)

49.9}}

1 + 134eº1.96t

E X A M P L E 4

GOAL 2

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Biology

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Botany

t 0 1 2 3 4 5 6 7 8 9 10

h 18 33 56 90 130 170 203 225 239 247 251

TUUV UW U V W X Y Z U [49.9

1 1 134e21.96 tA 5

Time (days)A

rea

(c

m2 )

A 5 49.9

(2.5, 25)

(0, 0.37)

Bacteria Growth

X=5 Y=170.21

KEYSTROKE HELP

Visit our Web sitewww.mcdougallittell.comto see keystrokes forseveral models ofcalculators.

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\ ] ^ _ ` a b c


1. What is the name of a function having the form y = where c, a, and rare positive constants?

2. What is a significant difference between using exponential growth functions andusing logistic growth functions as models for real-life quantities?

3. What is the significance of the point (ln 3, 4) on the graph of ƒ(x) = ?

Evaluate the function ƒ(x) = for the given value of x.

4. ƒ(0) 5. ƒ(º2) 6. ƒ(5) 7. ƒSº D 8. ƒ(10)

Graph the function. Identify the asymptotes, y-intercept, and point of

maximum growth.

9. ƒ(x) = 10. ƒ(x) = 11. ƒ(x) =

Solve the equation.

12. = 10 13. = 10 14. = 9

15. PLANTING SEEDS You planted a seedling and kept track of its height h (incentimeters) over time t (in weeks). Use the data in the table to find a model

that gives h as a function of t.

EVALUATING FUNCTIONS Evaluate the function ƒ(x) = for the

given value of x.

16. ƒ(1) 17. ƒ(3) 18. ƒ(º1) 19. ƒ(º6)

20. ƒ(0) 21. ƒS}

34

}D 22. ƒ(2.2) 23. ƒ(º0.9)


24. ƒ(x) = 25. ƒ(x) = 26. ƒ(x) =

A. B. C. d eZVd eZVd eZV 2}}

1 + 3eº4x3

}}

1 + 2eº4x4

}}

1 + 2eº3x

7}}

1 + 3eºx


12.5}}

1 + 7eº0.2x30

}}

1 + 4eºx18

}}

1 + 2eº2x

2}}

1 + 4eº0.25x8

}}

1 + 3eº0.4x5

}}

1 + 4eº2.5x

1}

2

12}}

1 + 5eº2x

8}

1 + 3e–x

c}}

1 + aeºrx

GUIDED PRACTICE

Concept Check ✓

Skill Check ✓


STUDENT HELP

HOMEWORK HELP

Example 1: Exs. 16–23Example 2: Exs. 24–35Example 3: Exs. 36–44Example 4: Exs. 45–49Example 5: Exs. 50, 51


STUDENT HELP

t 0 1 2 3 4 5 6 7 8

h 5 12 26 39 51 88 94 103 112

\ ] ^ _ ` a b c

t 0 1 2 3 4 5 6 7 8 9

y 9.6 18.3 29.0 47.2 71.1 119.1 174.6 257.3 350.7 441.0


GRAPHING FUNCTIONS Graph the function. Identify the asymptotes,

y-intercept, and point of maximum growth.

27. y = 28. y = 29. y =

30. y = 31. y = 32. y =

33. y = 34. y = 35. y =

SOLVING EQUATIONS Solve the equation.

36. = 5 37. = 9 38. = 1

39. = 20 40. = 68 41. = 30

42. = 7 43. = 44. = 6.4

OWNING A VCR In Exercises 45–47, use the following information.

The number of households in the United States that own VCRs has shown logistic

growth from 1980 through 1999. The number H (in millions) of households can be

modeled by the equationH =

where t is the number of years since 1980. c Source: Veronis, Suhler & Associates

45. In what year were there approximately 86 million households with VCRs?

46. Graph the model. In what year did the growth rate for the number of householdsstop increasing and start decreasing?

47. What is the long-term trend in VCR ownership?

ECONOMICS In Exercises 48 and 49, use the following information.

The gross domestic product (GDP) of the United States has shown logistic growthfrom 1970 through 1992. The gross domestic product G (in billions of dollars) can bemodeled by the equation

G =

where t is the number of years since 1970. c Source: U.S. Bureau of the Census

48. In what year was the GDP approximately $5000 billion?

49. Graph the model. When did the GDP reach its point of maximum growth?

YEAST POPULATION In Exercises 50 and 51, use the following information.

In biology class, you observed the biomass of a yeast population over a period oftime. The table gives the yeast mass y (in grams) after t hours.

50. Draw a scatter plot of the data.

51. Find a model that gives y as a function of t using the logistic regression feature ofa graphing calculator.

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1 + 13eº2x

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1 + 2eº4x8

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1 + 3eºx

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1 + 6eº0.5x8

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1 + eº1.02x

3}}

1 + 3eº8x4

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1 + 3eº3x4

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1 + 0.08eº2.1x

5}}

1 + eº10x2

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1 + 0.4eº0.3x1

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1 + 6eºx

ECONOMICS

Gross domesticproduct, the focus of Exs. 48and 49, is the value of allgoods and servicesproduced within a countryduring a given period.

RE

AL LIFE

RE

AL LIFE

FOCUS ON

APPLICATIONS

f g h i j k l m


52. MULTI-STEP PROBLEM The table shows the population P (in millions) of the United States from 1800 to 1870 where t represents the number of

years since 1800. c Source: U.S. Bureau of the Census

a. Use a graphing calculator to find an exponential growthmodel and a logistic growth model for the data. Then graph both models.

b. Use the models from part (a) to find the year when the population was about 92 million. Which of the models givesa year that is closer to 1910, the correct answer? Explainwhy you think that model is more accurate.

c. Use each model to predict the population in 2010. Which model gives a population closer to 297.7 million, thepredicted population from the U.S. Bureau of the Census?

53. ANALYZING MODELS The graph of a logistic growth function

y = reaches its point of maximum growth where y = .

Show that the x-coordinate of this point is x = .

WRITING EQUATIONS The variables x and y vary directly. Write an equation

that relates the variables. (Review 2.4 for 9.1)

54. x = 4, y = 36 55. x = º5, y = 10 56. x = 2, y = 13

57. x = 40, y = 5 58. x = 0.1, y = 0.9 59. x = 1, y = 0.2

WRITING EQUATIONS Write y as a function of x. (Review 8.7)

60. log y = 0.9 log x + 2.11 61. ln y = 0.94 º 2.44x

62. log y = º1.82 + 0.4x 63. log y = º0.75 log x º 1.76

Write an exponential function of the form y = abx whose graph passes through

the given points. (Lesson 8.7)

1. (2, 3), (5, 12) 2. (1, 16), (3, 45) 3. (5, 9), (8, 35)

Write a power function of the form y = ax b whose graph passes through the

given points. (Lesson 8.7)

4. (2, 28), (8, 192) 5. (1, 0.5), (6, 48) 6. (5, 40), (2, 6)

7. FLU VIRUS The spread of a virus through a student population can be

modeled by S = where S is the total number of students infected

after t days. Graph the model and tell when the point of maximum growth ininfections is reached. (Lesson 8.8)

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1 + 4999eº0.8t

QUIZ 3 Self-Test for Lessons 8.7 and 8.8

MIXED REVIEW

ln a}

r

c}

2c

}}

1 + aeºrx

TestPreparation

★★Challenge

DATA UPDATE

Visit our Web sitewww.mcdougallittell.com

INT

ERNET

STUDENT HELP

t P

0 5.3

10 7.2

20 9.6

30 12.9

40 17.0

50 23.2

60 31.4

70 39.8

n o p q r s t u

523

Chapter SummaryCHAPTER

8

What did you learn?

Graph exponential functions.

• exponential growth functions (8.1)

• exponential decay functions (8.2)

• natural base functions (8.3)

Evaluate and simplify expressions.

• exponential expressions with base e (8.3)

• logarithmic expressions (8.4)

Graph logarithmic functions. (8.4)

Use properties of logarithms. (8.5)

Solve exponential and logarithmic equations. (8.6)

Model data with exponential and power functions.(8.7)

Evaluate and graph logistic growth functions. (8.8)

Use exponential, logarithmic, and logistic growth functions to model real-life situations. (8.1–8.8)

How does Chapter 8 fit into the BIGGER PICTURE of algebra?

In Chapter 2 you began your study of functions and learned that quantities thatincrease by the same amount over equal periods of time are modeled by linearfunctions. In Chapter 8 you saw that quantities that increase by the samepercent over equal periods of time are modeled by exponential functions.

Exponential functions and logarithmic functions are two important “families”of functions. They model many real-life situations, and they are used inadvanced mathematics topics such as calculus and probability.

How did you studywith a group?

Here is an example of a summaryprepared for Lesson 8.4 andpresented to the group, followingthe Study Strategy on page 464.

STUDY STRATEGY

Study Group

Lesson 8.4 SummaryDefinition of logarithm: logb y = x if and only if b x

= yCommon logarithm (base 10): log10 x = log xNatural logarithm (base e): loge x = ln xInverse functions: ƒ(x) = b x (exponential) and

g(x) = logb x (logarithmic)Graph of logarithmic function ƒ(x) = logb (x º h) + k:asymptote x = h; domain x > h; range all realnumbers; up b > 0; down 0 < b < 1

Why did you learn it?

Estimate wind energy generated by turbines. (p. 470)

Find the depreciated value of a car. (p. 476)

Find the number of endangered species. (p. 482)

Find air pressure on Mount Everest. (p. 484)

Approximate distance traveled by a tornado. (p. 491)

Estimate the average diameter of sand particles for abeach with given slope. (p. 489)

Compare loudness of sounds. (p. 495)

Use Newton’s law of cooling. (p. 502)

Model the number of U.S. stamps issued. (p. 515)

Model the height of a sunflower. (p. 519)

Model a telescope’s limiting magnitude. (p. 507)

n o p q r s t u


Chapter ReviewCHAPTER

8

• exponential function, p. 465

• base of an exponential function, p. 465

• asymptote, p. 465

• exponential growth function, p. 466

• growth factor, p. 467

• exponential decay function, p. 474

• decay factor, p. 476

• natural base e, or Euler number, p. 480

• logarithm of y with base b, p. 486

• common logarithm, p. 487

• natural logarithm, p. 487

• change-of-base formula, p. 494

• logistic growth function, p. 517

VOCABULARY

8.1 EXPONENTIAL GROWTH

An exponential growth function has the form y = abx with a > 0 and b > 1.

To graph y = 2 • 5x + 2 º 4, first lightly sketch the graph of y = 2 • 5x, which passes through (0, 2) and (1, 10). Then translate the graph 2 units to the left and 4 units down. The graph passes through (º2, º2) and (º1, 6). The asymptote is the line y = º4. The domain is all real numbers,and the range is y > º4.

Examples onpp. 465–468


1. y = º2x + 4 2. y = 3 • 2x 3. y = 5 • 3x º 2 4. y = 4x + 3 º 1

8.2 EXPONENTIAL DECAY

An exponential decay function has the form y = abx witha > 0 and 0 < b < 1.

To graph y = 4S}

13

}Dx, plot (0, 4) and S1, }

43

}D. From right to left draw a

curve that begins just above the x-axis, passes through the two points, and moves up. The asymptote is the line y = 0. The domain is all realnumbers, and the range is y > 0.


Tell whether the function represents exponential growth or exponential decay.

5. ƒ(x) = 5S}

34

}Dx6. ƒ(x) = 2S}

54

}Dx7. ƒ(x) = 3(6)ºx 8. ƒ(x) = 4(3)x


9. y = S}

14

}Dx10. y = 2S}

35

}Dx º 111. y = S}

12

}Dxº 5 12. y = º3S}

34

}Dx+ 2

EXAMPLE

EXAMPLE

v2

w xy 5 2 d 5x

y 5 2 d 5 x 1 2 2 4

xyy y 5 4x cx13

z

n o p q r s t u

Chapter Review 525

8.3 THE NUMBER e

You can use e as the base of an exponential function.To graph such a function, use e ≈ 2.718 and plot some points.

ƒ(x) = 3e2x is an exponential growth function, since 2 > 0.g(x) = 3eº2x is an exponential decay function, since º2 < 0.

For both functions, the y-intercept is 3, the asymptote is y = 0, the domain is all real numbers, and the range is y > 0.



13. y = ex + 5 14. y = 0.4ex º 3 15. y = 4eº2x 16. y = ºex + 3

8.4 LOGARITHMIC FUNCTIONS

You can use the definition of logarithm to evaluate expressions: logb y = x if and only if bx = y. The common logarithm has base 10 (log10 x = log x).The natural logarithm has base e (loge x = ln x).

To evaluate log8 4096, write log8 4096 = log8 84 = 4.

To graph the logarithmic function ƒ(x) = 2 log x + 1, plot points such as (1, 1) and (10, 3). The vertical line x = 0 is an asymptote. The domain is x > 0, and the range is all real numbers.


Evaluate the expression without using a calculator.

17. log4 64 18. log2 }

18

} 19. log3 }

19

} 20. log6 1


21. y = 3 log5 x 22. y = log 4x 23. y = ln x + 4 24. y = log (x º 2)

8.5 PROPERTIES OF LOGARITHMS

You can use product, quotient, and power properties of logarithms.

Expand: log2 }

3yx} = log2 3x º log2 y = log2 3 + log2 x º log2 y

Condense: 3 log6 4 + log6 2 = log6 43 + log6 2 = log6 (64 • 2) = log6 128


Expand the expression.

25. log3 6xy 26. ln }

73x} 27. log 5x3 28. log }

x5

2

y

y

º2

}

Condense the expression.

29. 2 ln 3 º ln 5 30. log4 3 + 3 log4 2 31. 0.5 log 4 + 2(log 6 º log 2)

EXAMPLES

EXAMPLES

EXAMPLES

{|}y 5 3e 2x y 5 3e22x

~ {||ƒ(x) 5 2 log x 1 1

� � � � � � � �8.6 SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS


You can solve exponential equations by equating exponents or by takingthe logarithm of each side. You can solve logarithmic equations by exponentiating eachside of the equation.

10x = 4.3 log4 x = 3

log 10x = log 4.3 Take log of each side. 4log4 x = 43 Exponentiate each side.

x = log 4.3 ≈ 0.633 x = 43 = 64

EXAMPLES

8.8 LOGISTIC GROWTH FUNCTIONSExamples onpp. 517–519

You can graph logistic growth functions by plotting points and identifying important characteristics of the graph.

The graph of y = is shown. It has asymptotes

y = 0 and y = 6. The y-intercept is 1.5. The point of maximum

growth is S}

ln2

3}, }

62

}D ≈ (0.55, 3).

6}}

1 + 3eº2x

EXAMPLE

8.7 MODELING WITH EXPONENTIAL AND POWER FUNCTIONS

You can write an exponential function of the form y = abx or a powerfunction of the form y = axb that passes through two given points.

To find a power function given (3, 2) and (9, 12), substitute the coordinates into y = axb to get the equations 2 = a • 3b and 12 = a • 9b. Solve the system of equationsby substitution: a ≈ 0.333 and b ≈ 1.631. So, the function is y = 0.333x1.631.


EXAMPLE

Solve the equation. Check for extraneous solutions.

32. 2(3)2x = 5 33. 3eºx º 4 = 9 34. 3 + ln x = 8 35. 5 log (x º 2) = 11

Find an exponential function of the form y = abx whose graph passes through

the given points.

36. (2, 6), (3, 8) 37. (2, 8.9), (4, 20) 38. (2, 4.2), (4, 3.6)

Find a power function of the form y = axb whose graph passes through the

given points.

39. (2, 3.4), (6, 7.3) 40. (2, 12.5), (4, 33.2) 41. (0.5, 1), (10, 150)

~{|� (0.55, 3)

Graph the function. Identify the asymptotes, y-intercept, and point of maximum

growth.

42. y = 43. y = 44. y = 3

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1 + 0.5eº0.5x

4}}

1 + 2eº3x

2}

1 + eº2x


� � � � � � � �Graph the function. State the domain and range.

1. y = 2S}

16

}Dx2. y = 4x º 2 º 1 3. y = }

12

}ex + 1 4. y = eº0.4x

5. y = log1/2 x 6. y = ln x º 4 7. y = log (x + 6) 8. y =


9. (2eº1)(3e2) 10. 11. e6 • ex • eº3x 12. log 10002 13. 8log8 x

Evaluate the expression without using a calculator.

14. log4 0.25 15. log1/3 27 16. log 1 17. ln eº2 18. log3 2432

Solve the equation. Check for extraneous solutions.

19. 12 = 10x + 5 º 7 20. 5 º ln x = 7 21. log2 4x = log2 (x + 15) 22. = 3.3

23. Tell whether the function ƒ(x) = 10(0.87)x represents exponential growth orexponential decay.

24. Find the inverse of the function y = log6 x.

25. Use log2 5 ≈ 2.322 to approximate log2 50 and log2 0.4.

26. Condense the expression 3 log4 14 º 3 log4 42.

27. Expand the expression ln 2y2x.

28. Use the change-of-base formula to evaluate the expression log7 15.

29. Find an exponential function of the form y = abx whose graph passes throughthe points (4, 6) and (7, 10).

30. Find a power function of the form y = axb whose graph passes through thepoints (2, 3) and (10, 21).

31. CAR DEPRECIATION The value of a new car purchased for $24,900decreases by 10% per year. Write an exponential decay model for the value ofthe car. After about how many years will the car be worth half its purchase price?

32. EARNING INTEREST You deposit $4000 in an account that pays 7% annualinterest compounded continuously. Find the balance at the end of 5 years.

33. COD WEIGHT The table gives the mean weight w (in kilograms) and age x(in years) of Atlantic cod from the Gulf of Maine.

a. Draw a scatter plot of ln w versus x. Is an exponential model a good fit for theoriginal data?

b. Find an exponential model for the original data. Estimate the weight of a codthat is 9 years old.

4}}

1 + 2.5eº4x

º4ex

}

2e5x

2}

1 + 2eºx

Chapter Test 527

Chapter TestCHAPTER

8

x 1 2 3 4 5 6 7 8

w 0.751 1.079 1.702 2.198 3.438 4.347 7.071 11.518

� � � � � � � �


Chapter Standardized TestCHAPTER

8

1. MULTIPLE CHOICE Which function is graphed?

¡A ƒ(x) = 3(0.8)x

¡B ƒ(x) = 2(0.8)x º 3

¡C ƒ(x) = 2(0.8)x º 3

¡D ƒ(x) = 4(0.8)x º 3

¡E ƒ(x) = 4(0.8)x º 3

2. MULTIPLE CHOICE Suppose you deposit money inan investment account that pays 7% annual interestcompounded continuously. About how many yearswill it take for your initial deposit to double?

¡A 5 ¡B 7 ¡C 9

¡D 10 ¡E 14

3. MULTIPLE CHOICE Which function is the inverseof y = ln (x º 2)?

¡A y = ex + 2 ¡B y = ex + 2

¡C y = ex º 2 ¡D y = ex º 2

¡E y = eº2x

4. MULTIPLE CHOICE Which function is graphed?

¡A ƒ(x) = log 4x

¡B ƒ(x) = 4 + log x

¡C ƒ(x) = º4 + log x

¡D ƒ(x) = log (x + 4)

¡E ƒ(x) = log (x º 4)

5. MULTIPLE CHOICE Which of the following isequivalent to log2 7?

¡A 72¡B 27

¡C 7 log 2

¡D }

l

l

o

o

g

g

7

2} ¡E }

l

l

o

o

g

g

7

2}

6. MULTIPLE CHOICE Which of the following is

equivalent to log }

x

z

y2

}?

¡A log x + 2 log y + log z

¡B log x + 2 log y º log z

¡C log z º log x º 2 log y

¡D 2 log xy º log z

¡E log z º 2 log xy

7. MULTIPLE CHOICE What is the solution of theequation 2x + 14 = 162x?

¡A 2 ¡B 4 ¡C 8

¡D 16 ¡E No solution

8. MULTIPLE CHOICE What is the solution of theequation 0.5 log3 x = 2?

¡A 4 ¡B 64 ¡C 81

¡D }

614} ¡E }

811}

9. MULTIPLE CHOICE Which function does not have agraph with asymptote y = 0?

¡A ƒ(x) = 3x¡B ƒ(x) = (0.25)x + 2

¡C ƒ(x) = log 6x ¡D ƒ(x) = eº5x

¡E ƒ(x) =

10. MULTIPLE CHOICE What type of function is ƒ(x) = 4e

0.5x?

¡A Exponential decay function

¡B Exponential growth function

¡C Logarithmic function

¡D Logistic growth function

¡E Power function

1}

1 + eº2x

(23, 0)

� ��

� ��(0, 1)

TEST-TAKING STRATEGY If you get stuck on a question, look at the answer choices for clues.

Or select an answer choice and check to see if it is a reasonable answer to the question.

� � � � � � � �QUANTITATIVE COMPARISON In Exercises 11 and 12, choose the statement

that is true about the given quantities.

¡A The quantity in column A is greater.

¡B The quantity in column B is greater.

¡C The two quantities are equal.

¡D The relationship cannot be determined from the given information.

11.

12.

13. MULTI-STEP PROBLEM You are considering two job offers. The first offer is asalary of $32,000 with a $550 annual raise. The other offer is a salary of $29,500with a 4% annual raise.

a. Write a linear model for the total salary with the first offer as a function of thenumber t of years.

b. Write an exponential model for the total salary with the other offer as afunction of the number t of years.

c. Graph the functions in the same coordinate plane with domain 0 ≤ t ≤ 8.Find the point of intersection of the two graphs and tell what it represents.

d. Writing Explain the difference in the salaries over time.

14. MULTI-STEP PROBLEM The table gives the weight w (in pounds) of an average girlfor the first five years of life where t is her age in months. c Source: Your Baby & Child

a. Draw a scatter plot of ln w versus t.

b. Draw a scatter plot of ln w versus ln t.

c. Analyze your scatter plots and decide whether an exponential model or apower model is a better fit for the original data. Explain your choice.

d. Using your answer from part (c), find a model for the data. Check your modelby using the regression feature of a graphing calculator.

e. Use your model to estimate a girl’s weight at 1}

12

}, 2}

12

}, 3}

12

}, and 4}

12

} years old.

15. MULTI-STEP PROBLEM Use the function ƒ(x) = .

a. Find ƒ(º1), ƒ(0), and ƒ(2).

b. Sketch a graph of the function.

c. Identify the asymptotes, y-intercept, and point of maximum growth.

d. Write and solve an equation to find the value of x when ƒ(x) equals 4. Labelthis point on your graph.

e. Writing Describe how the growth represented by this function changes over time.

5}

1 + 9eºx

Chapter Standardized Test 529

Column A Column B

log 10,000 ln e4

log2 4 log4 2

t 2 4 6 8 10 12 24 36 48 60

w 10.5 13.5 16.0 18.5 20.0 21.5 27.5 31.5 35.5 39.0

EXPONENTIAL AND LOGARITHMIC...

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