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D. Finding the Domain of a Logarithmic Function

D. Finding the Domain of a Logarithmic Function

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3Ϫx

3Ϫx

7 0, we note y ϭ

has a zero at x ϭ 3, with a vertical

xϩ3

xϩ3

asymptote at x ϭ Ϫ3 and here we opt to use the interval test method to solve the

inequality. Outputs are positive when x ϭ 0 (see Figure 5.30), so y is positive in

the interval (Ϫ3, 3) and negative elsewhere. The domain of r is x ʦ 1Ϫ3, 32 .

c. For

Figure 5.30

When Ϫ3 Ͻ x Ͻ 3,

yϾ0

(interval test)

Ϫ4

Ϫ3

Ϫ2

Ϫ1

0

2

1

3

When x Ͻ Ϫ3, y Ͻ 0

D. You’ve just seen how

we can find the domain of a

logarithmic function

4

x

When x Ͼ 3, y Ͻ 0

d. For Ϳx Ϫ 2Ϳ 7 0, we note y ϭ Ϳx Ϫ 2Ϳ is the graph of y ϭ ͿxͿ shifted 2 units

right, with its vertex at (2, 0). The graph is positive for all x, except at x ϭ 2.

The domain of f is x ʦ 1Ϫq, 22 ´ 12, q 2 .

Now try Exercises 73 through 78 ᮣ

3Ϫx

b from

xϩ3

Example 7c can also be confirmed using the

LOG

key on a graphing calculator. Use this

key to enter the equation as Y1 on the Y=

screen, then graph the function using the

ZOOM 4:ZDecimal option. Both the graph (Figure 5.31) and TABLE feature help to confirm

the domain is x ʦ 1Ϫ3, 32 .

Figure 5.31

The domain for r 1x2 ϭ log a

3.1

Ϫ4.7

4.7

Ϫ3.1

E. Applications of Logarithms

The use of logarithmic scales as a tool of measurement is primarily due to the range of

values for the phenomenon being measured. For instance, time is generally measured on

a linear scale, and for short periods a linear scale is appropriate. For the time line in Figure 5.32, each tick-mark represents 1 unit, and the time line can display a period of 10 yr.

However, the scale would be useless in a study of geology or the age of the universe. If

we scale the number line logarithmically, each tick-mark represents a power of 10 (Figure 5.33) and a scale of the same length can now display a time period of 10 billion years.

Figure 5.32

years

0

1

2

3

4

5

6

7

8

9

10

Figure 5.33

0

(101) (102) (103) (104) (105) (106) (107) (108) (109) (1010)

years

0

WORTHY OF NOTE

The decibel (dB) is the reference

unit for sound, and is based on the

faintest sound a person can hear,

called the threshold of audibility. It

is a base-10 logarithmic scale,

meaning a sound 10 times more

intense is one bel louder.

1

2

3

4

5

6

7

8

9

10

In much the same way, logarithmic measures are needed in a study of sound and

earthquake intensity, as the scream of a jet engine is over 1 billion times more intense

than the threshold of hearing, and the most destructive earthquakes are billions of

times stronger than the slightest earth movement that can be felt. Similar ranges exist

in the measurement of light, acidity, and voltage. Figures 5.34 and 5.35 show logarithmic scales for measuring sound in decibels (1 bel ϭ 10 decibels) and earthquake

intensity in Richter values (or magnitudes).

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509

s

ci

ty

lo

u

co

nv

er

s

ft

so

at

io

n

h

ap um

pl of

ia an

nc

e

er

hi

sp

w

ol

d

es

h

th

r

tra

ffi

c

d

m

ot

or

je

cy

tf

cl

ly

e

-o

ve

r

Figure 5.34

8

9

bels

0

1

2

3

4

5

6

7

10

de

va

sta

ca ting

ta

str

op

hi

ne

c

ve

rr

ec

or

de

d

uc

tiv

e

de

str

n

sh otic

ak ea

in ble

g

fe

lt

ba

re

ly

th

r

es

h

ol

d

Figure 5.35

magnitudes

0

1

2

3

4

5

6

1992

San Jose,

CA (5.5)

7

8

9

10

1906

2004

San Fran, Indian

CA (8.1) Ocean (9.3)

The slightest earth movement perceptible is called the reference intensity I0, with the

intensity I of stronger earthquakes expressed as a multiple of I0. The earthquake that struck

Haiti in January of 2010 was measured at over 10,500,000 times this reference intensity, or

I ϭ 10,5000,000I0. To find the Richter value (magnitude) of this earthquake, we simply

I

take the base-10 logarithm of the ratio to express these values on a logarithmic scale. In

I0

I

function form, M1I2 ϭ log a b, and we find that the Haitian earthquake had a magnitude

I0

10,500,000I0

of just over 7.0: log a

b ϭ log 110,500,0002 Ϸ 7.0.

I0

EXAMPLE 7A

Finding the Magnitude of an Earthquake

Find the magnitude of the earthquakes (rounded to hundredths) with the intensities given.

a. Eureka earthquake; January 9, 2010, near Humboldt county, California:

I ϭ 3,162,000I0.

b. Sumatra-Andaman earthquake; December 26, 2004, near the west coast of

Sumatra, Indonesia: I ϭ 1,995,260,000I0.

Solution

I

M1I2 ϭ log a b

I0

3,162,000I0

b

M13,162,000I0 2 ϭ log a

I0

ϭ log 3,162,000

Ϸ 6.5

The earthquake had a magnitude of about 6.5.

I

b.

M1I2 ϭ log a b

I0

1,995,260,000I0

b

M11,995,260,000I0 2 ϭ log a

I0

ϭ log 1,995,260,000

Ϸ 9.3

The earthquake had a magnitude of about 9.3.

a.

magnitude equation

substitute 3,162,000I0 for I

simplify

result

magnitude equation

substitute 1,995,260,000I0 for I

simplify

result

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EXAMPLE 7B

Comparing Earthquake Intensity to the Reference Intensity

How many times more intense than the reference intensity I0 was the Peruvian

earthquake of June 23, 2001, with magnitude 8.4.

Solution

I

M1I2 ϭ log a b

I0

I

8.4 ϭ log a b

I0

I

108.4 ϭ a b

I0

I ϭ 108.4I0

I Ϸ 251,188,643I0

magnitude equation

substitute 8.4 for M(I )

exponential form

solve for I

108.4 Ϸ 251,188,643

The earthquake was over 251 million times more intense than the reference intensity.

EXAMPLE 7C

Comparing Earthquake Intensities

Referring to Example 7A, how many times more

intense was the Sumatra earthquake as compared

to the Eureka earthquake?

Solution

The Sumatra quake had a Richter value of 9.3,

with an intensity of 109.3. Similarly, the Eureka

quake was measured at 6.5 on the Richter

scale, with an intensity of 106.5. Using these

intensities, we find that the Sumatra quake was

109.3

ϭ 102.8 or about 631 times more intense than the Eureka quake.

106.5

Now try Exercises 81 through 94 ᮣ

A second application of logarithmic functions involves the relationship between

altitude and barometric pressure. The altitude or height above sea level can be determined

P0

by the formula H ϭ 130T ϩ 80002 lna b, where H is the altitude in meters for a

P

temperature T in degrees Celsius, P is the barometric pressure at a given altitude in

units called centimeters of mercury (cmHg), and P0 is the barometric pressure at sea

level: 76 cmHg.

EXAMPLE 8

Using Logarithms to Determine Altitude

Hikers at the summit of Mt. Shasta in northern California take a pressure reading

of 45.1 cmHg at a temperature of 9°C. How high is Mt. Shasta?

Solution

For this exercise, P0 ϭ 76, P ϭ 45.1, and T ϭ 9. The formula yields

H ϭ 130T ϩ 80002 ln a

P0

b

P

76

b

ϭ 330192 ϩ 8000 4 ln a

45.1

76

ϭ 8270 ln a

b

45.1

Ϸ 4316

given formula

substitute given values

simplify

result

Mt. Shasta is about 4316 m high.

Now try Exercises 95 through 98 ᮣ

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Our final application shows the versatility of logarithmic functions, and their value

as a real-world model. Large advertising agencies are well aware that after a new ad

campaign, sales will increase rapidly as more people become aware of the product.

Continued advertising will give the new product additional market share, but once the

“newness” wears off and the competition begins responding, sales tend to taper off—

regardless of any additional amount spent on ads. This phenomenon can be modeled

by the function

S1d2 ϭ k ϩ a ln d

where S(d) is the number of expected sales after d dollars are spent, and a and k are

constants related to product type and market size.

EXAMPLE 9

Using Logarithms for Marketing Strategies

y

Market research has shown that sales of the

MusicMaster, a new system for downloading and

playing music, can be approximated by the

equation S1d2 ϭ 2500 ϩ 250 ln d, where S(d) is the

number of sales after d thousand dollars is spent on

advertising. The graph of y ϭ S1d2 is shown.

a. What sales volume is expected if the

advertising budget is \$40,000?

b. If the company needs to sell 3500 units to

begin making a profit, how much should be

spent on advertising?

Solution

4500

4000

y ϭ 2500 ϩ 250ln x

3500

3000

2500

2000

x

10 20 30 40 50 60 70 80 90 100

a. For sales volume, we simply evaluate the function for d ϭ 40 (d in thousands):

S1d2 ϭ 2500 ϩ 250 ln d

S1402 ϭ 2500 ϩ 250 ln 40

Ϸ 2500 ϩ 922

ϭ 3422

given equation

substitute 40 for d

250 ln 40 Ϸ 922

Spending \$40,000 on advertising will generate approximately 3422 sales.

b. To find the advertising budget needed, we substitute number of sales and solve

for d.

S1d2 ϭ 2500 ϩ 250 ln d

3500 ϭ 2500 ϩ 250 ln d

1000 ϭ 250 ln d

4 ϭ ln d

e4 ϭ d

54.598 Ϸ d

E. You’ve just seen how

we can solve applications of

logarithmic functions

given equation

substitute 3500 for S (d)

subtract 2500

divide by 250

exponential form

e4 Ϸ 54.598

About \$54,600 should be spent in order to sell 3500 units.

Now try Exercises 99 and 100 ᮣ

From the graph of S(d) given in Example 10, it is apparent that while the number of

sales continues to grow as more money is spent, the rate of growth slows considerably

beyond \$50,000. In cases like this a numeric view of what’s happening can be more

meaningful. Here we’ll use the TABLE feature in an entirely new way to investigate the

number of sales gained for each additional \$1000 spent. For Y1 ϭ 2500 ϩ 250 ln x,

we’ll enter Y2 ϭ Y1 1x2 Ϫ Y1 1x Ϫ 12 , which will automatically have the calculator find

the difference between the current number of sales (spending x thousand dollars), and

the number of sales made when \$1000 less is spent 1x Ϫ 1 dollars2 . Figure 5.36 shows

that initially each additional \$1000 spent results in a substantial sales increase, while

Figure 5.37 shows very minor increases for a like amount spent.

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Figure 5.36

Figure 5.37

There are a number of other interesting applications of logarithmic functions in the

Exercise set. See Exercises 101 through 106.

5.3 EXERCISES

CONCEPTS AND VOCABULARY

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.

1. A logarithmic function is of the form y ϭ

7 0,

1 and inputs are

where

than zero.

3. For logarithmic functions of the form y ϭ logb x,

.

the x-intercept is

, since logb1 ϭ

5. What number does the expression log232

represent? Discuss/Explain how log232 ϭ log225

justifies this fact.

,

2. The range of y ϭ logb x is all

, and

the domain is x ʦ

. Further, as x S 0,

yS

.

4. The function y ϭ logb x is an increasing function if

, and a decreasing function if

.

6. Explain how the graph of Y ϭ logb 1x Ϫ 32 can be

obtained from y ϭ logb x. Where is the “new”

x-intercept? Where is the new asymptote?

DEVELOPING YOUR SKILLS

Write each equation in exponential form.

7. 3 ϭ log28

9. Ϫ1 ϭ log7 17

11. 0 ϭ log91

13.

1

3

ϭ log82

8. 2 ϭ log39

10. Ϫ3 ϭ ln

1

e3

12. 0 ϭ ln 1

14.

1

2

ϭ log819

15. 1 ϭ log22

16. 1 ϭ ln e

17. log749 ϭ 2

18. log416 ϭ 2

19. log 100 ϭ 2

20. log 10,000 ϭ 4

21. ln 154.5982 Ϸ 4

22. log 0.001 ϭ Ϫ3

Write each equation in logarithmic form.

23. 43 ϭ 64

24. e3 Ϸ 20.086

25. 3Ϫ2 ϭ 19

26. 2Ϫ3 ϭ 18

27. e0 ϭ 1

28. 80 ϭ 1

31. 103 ϭ 1000

32. e1 ϭ e

29. 1 13 2 Ϫ3 ϭ 27

1

33. 10Ϫ2 ϭ 100

3

35. 42 ϭ 8

Ϫ3

37. 4 2 ϭ

1

8

30. 1 15 2 Ϫ2 ϭ 25

34. 10Ϫ5 ϭ

1

100,000

3

36. e4 Ϸ 2.117

Ϫ2

38. 27 3 ϭ 19

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Determine the value of each logarithm without using a

calculator.

39. log44

40. log99

41. log11121

42. log12144

43. ln e

44. ln e2

45. log4 2

46. log81 9

47.

513

Section 5.3 Logarithms and Logarithmic Functions

1

log7 49

1

49. ln 2

e

1

48. log9 81

51. log 50

52. log 47

53. ln 1.6

54. ln 0.75

55. ln 225

56. ln 381

58. log 4␲

Graph each function using transformations of y ‫ ؍‬logb x

and strategically plotting a few points. Clearly state the

transformations applied.

59. f 1x2 ϭ log2x ϩ 3

60. g1x2 ϭ log2 1x Ϫ 22

63. q1x2 ϭ ln1x ϩ 12

64. r 1x2 ϭ ln1x ϩ 12 Ϫ 2

II.

y

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

5

4

3

2

1

1 2 3 4 5 x

III.

1

50. ln

1e

Use a calculator to evaluate each expression, rounded to

four decimal places.

57. log 137

I.

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

IV.

y

y

5

4

3

2

1

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

V.

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

VI.

y

y

5

4

3

2

1

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

61. h1x2 ϭ log2 1x Ϫ 22 ϩ 3 62. p1x2 ϭ log3x Ϫ 2

65. Y1 ϭ Ϫln1x ϩ 12

66. Y2 ϭ Ϫln x ϩ 2

Use the transformation equation y ‫ ؍‬af 1x ؎ h2 ؎ k

and the asymptotes and intercept(s) of the parent

function to match each equation to one of the graphs

given. Assume b Ͼ 1.

67. y ϭ logb 1x ϩ 22

68. y ϭ 2logb x

69. y ϭ 1 Ϫ logb x

70. y ϭ logb x Ϫ 1

71. y ϭ logb x ϩ 2

72. y ϭ Ϫlogb x

Determine the domain of the following functions.

73. y ϭ log6 a

xϩ1

b

xϪ3

75. y ϭ log5 12x Ϫ 3

77. y ϭ log19 Ϫ x 2

2

74. y ϭ ln a

xϪ2

b

xϩ3

76. y ϭ ln 15 Ϫ 3x

78. y ϭ ln19x Ϫ x2 2

WORKING THE FORMULAS

79. pH level: f 1x2 ‫ ؍‬؊log x

The pH level of a solution indicates the concentration

of hydrogen 1H ϩ 2 ions in a unit called moles per

liter. The pH level f(x) is given by the formula shown

(often written as pH ϭ Ϫlog[Hϩ]), where x is the

ion concentration (given in scientific notation). A

solution with pH 6 7 is called an acid (lemon juice:

pH Ϸ 22, and a solution with pH 7 7 is called a

base (household ammonia: pH Ϸ 112. Use the

formula to determine the pH level of tomato juice if

x ϭ 7.94 ϫ 10Ϫ5 moles per liter. Is this an acid or

base solution?

80. Time required for an investment to double:

log 2

T1r2 ‫؍‬

log11 ؉ r2

The time required for an investment to double in

value is given by the formula shown, where T(r)

represents the time required for an investment to

double if invested at interest rate r (expressed as a

decimal). How long would it take an investment to

double if the interest rate were (a) 5%, (b) 8%,

(c) 12%?

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APPLICATIONS

Earthquake intensity: Use the information provided in

Example 8 to answer the following.

81. Find the value of M(I) given

a. I ϭ 50,000I0

b. I ϭ 75,000,000 I0.

82. Find the intensity I of the earthquake given

a. M1I2 ϭ 3.2

b. M1I2 ϭ 8.1.

Determine how many times more intense the first quake

was compared to the second.

83. Great Chilean quake (1960): magnitude 9.5

Kobe, Japan, quake (1995): magnitude 6.9

84. Northern Sumatra (2004): magnitude 9.1

Southern Greece (2008): magnitude 4.5

85. Earthquake intensity: On June 25, 1989, an

earthquake with magnitude 6.2 shook the southeast

side of the Island of Hawaii (near Kalapana),

causing some \$1,000,000 in damage. On October

15, 2006, an earthquake measuring 6.7 on the

Richter scale shook the northwest side of the

island, causing over \$100,000,000 in damage. How

much more intense was the 2006 quake?

86. Earthquake intensity: The most intense

earthquake of the modern era occurred in Chile on

May 22, 1960, and measured 9.5 on the Richter

scale. How many times more intense was this

earthquake, than the quake that hit Northern

Sumatra (Indonesia) on March 28, 2005, and

measured 8.7?

Brightness of a star: The brightness or intensity I

of a star as perceived by the naked eye is measured

in units called magnitudes. The brightest stars have

magnitude 1 3 M1I2 ϭ 14 and the dimmest have

magnitude 6 3 M1I2 ϭ 6 4. The magnitude of a star

I

is given by the equation M1I2 ϭ 6 Ϫ 2.5 # log a b,

I0

where I is the actual intensity of light from the

star and I0 is the faintest light visible to the human

eye, called the reference intensity. The intensity

I is often given as a multiple of this reference

intensity.

87. Find the value of M(I) given

a. I ϭ 27I0 and b. I ϭ 85I0.

88. Find the intensity I of a star given

a. M1I2 ϭ 1.6 and b. M1I2 ϭ 5.2.

Intensity of sound: The intensity of sound as

perceived by the human ear is measured in units

called decibels (dB). The loudest sounds that can

be withstood without damage to the eardrum are in

the 120- to 130-dB range, while a whisper may

measure in the 15- to 20-dB range. Decibel measure

I

is given by the equation D1I2 ϭ 10 log a b, where

I0

I is the actual intensity of the sound and I0 is the

faintest sound perceptible by the human ear—called the

reference intensity. The intensity I is often given as a

multiple of this reference intensity, with the constant

10Ϫ16 (watts per cm2; W/cm2) used as the threshold

of audibility.

89. Find the value of D(I) given

a. I ϭ 10Ϫ14 and b. I ϭ 10Ϫ4.

90. Find the intensity I of the sound given

a. D1I2 ϭ 83 and b. D1I2 ϭ 125.

Determine how many times more intense the first sound

is compared to the second.

91. pneumatic hammer: 11.2 bels

heavy lawn mower: 8.5 bels

92. train horn: 7.5 bels

soft music: 3.4 bels

93. Sound intensity of a hair dryer: Every morning

(it seems), Jose is awakened by the mind-jarring,

ear-jamming sound of his daughter’s hair dryer

(75 dB). He knew he was exaggerating, but told

her (many times) of how it reminded him of his

railroad days, when the air compressor for the

pneumatic tools was running (110 dB). In fact, how

many times more intense was the sound of the air

compressor compared to the sound of the hair

dryer?

94. Sound intensity of a busy street: The decibel level

of noisy, downtown traffic has been estimated at

87 dB, while the laughter and banter at a loud party

might be in the 60 dB range. How many times more

intense is the sound of the downtown traffic?

P0

The barometric equation H ‫ ؍‬130T ؉ 80002 lna b was

P

discussed in Example 9.

95. Temperature and atmospheric pressure:

Determine the height of Mount McKinley (Alaska),

if the temperature at the summit is Ϫ10°C, with a

barometric reading of 34 cmHg.

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96. Temperature and atmospheric pressure: A

large passenger plane is flying cross-country. The

instruments on board show an air temperature of

3°C, with a barometric pressure of 22 cmHg. What

is the altitude of the plane?

97. Altitude and atmospheric pressure: By

definition, a mountain pass is a low point between

two mountains. Passes may be very short with

steep slopes, or as large as a valley between two

peaks. Perhaps the highest drivable pass in the

world is the Semo La pass in central Tibet. At its

highest elevation, a temperature reading of 8°C

was taken, along with a barometer reading of

39.3 cmHg. (a) Approximately how high is the

Semo La pass? (b) While traveling up to this pass,

an elevation marker is seen. If the barometer

reading was 47.1 cmHg at a temperature of 12°C,

what height did the marker give?

98. Altitude and atmospheric pressure: Hikers on

Mt. Everest take successive readings of 35 cmHg at

5°C and 30 cmHg at Ϫ10°C. (a) How far up the

mountain are they at each reading? (b) Approximate

the height of Mt. Everest if the temperature at the

summit is Ϫ27°C and the barometric pressure is

22.2 cmHg.

99. Marketing budgets: An advertising agency has

determined the number of items sold by a certain

client is modeled by the equation

N1A2 ϭ 1500 ϩ 315 ln A, where N(A) represents

the number of sales after spending A thousands of

dollars on advertising. Determine the approximate

number of items sold on an advertising budget of

(a) \$10,000; (b) \$50,000. (c) Use the TABLE

feature of a calculator to estimate how large a

budget is needed (to the nearest \$500 dollars) to

sell 3000 items.

100. Sports promotions: The accountants for a major

boxing promoter have determined that the number

of pay-per-view subscriptions sold to their

championship bouts can be modeled by the

function N1d2 ϭ 15,000 ϩ 5850 ln d, where N(d)

represents the number of subscriptions sold after

spending d thousand dollars on promotional

activities. Determine the number of subscriptions

sold if (a) \$50,000 and (b) \$100,000 is spent.

(c) Determine how much should be spent (to the

nearest \$1000 dollars) to sell over 50,000

subscriptions by simplifying the logarithmic

equation and writing the result in exponential form.

101. Home ventilation: In the construction of new

housing, there is considerable emphasis placed on

correct ventilation. If too little outdoor air enters a

home, pollutants can sometimes accumulate to

levels that pose a health risk. For homes of various

sizes, ventilation requirements have been

Section 5.3 Logarithms and Logarithmic Functions

515

established and are based on floor area and the

number of bedrooms. For a three-bedroom home,

the relationship can be modeled by the function

C1x2 ϭ 42 ln x Ϫ 270, where C(x) represents the

number of cubic feet of air per minute (cfm) that

should be exchanged with outside air in a home with

floor area x (in square feet). (a) How many cfm of

exchanged air are needed for a three-bedroom home

with a floor area of 2500 ft2? (b) If a three-bedroom

home is being mechanically ventilated by a system

with 40 cfm capacity, what is the square footage of

the home, assuming it is built to code?

102. Runway takeoff

distance: Many

will remember the

August 27, 2006,

crash of a

commuter jet at

Lexington’s Blue

Grass Airport, that

was mistakenly trying to take off on a runway that

was just too short. Forty-nine lives were lost. The

minimum required length of a runway depends on

the maximum allowable takeoff weight (mtw) of a

specific plane. This relationship can be approximated

by the function L1x2 ϭ 2085 ln x Ϫ 14,900, where

L(x) represents the required length of a runway in

feet, for a plane with x mtw in pounds.

a. The Airbus-320 has a 169,750 lb mtw. What

minimum runway length is required for takeoff?

b. By simplifying the logarithmic equation that

results and writing the equation in exponential

form, determine the mtw of a Learjet 30, which

requires a runway of 5550 ft to takeoff safely.

Memory retention: Under certain conditions, a person’s

retention of random facts can be modeled by the equation

P1x2 ϭ 95 Ϫ 14 log2x, where P(x) is the percentage of

those facts retained after x number of days. Find the

percentage of facts a person might retain after x days for the

values given. Note that many of the values given are powers

of 2. Use the change-of-base formula those that are not.

103. a. 1 day

b. 4 days

c. 16 days

104. a. 32 days

b. 64 days

c. 78 days

105. pH level: Use the formula given in Exercise 79 to

determine the pH level of black coffee if

x ϭ 5.1 ϫ 10Ϫ5 moles per liter. Is black coffee

considered an acid or base solution?

106. Tripling time: The length of time required for an

amount of money to triple is given by the formula

log 3

T1r2 ϭ

(see Exercise 80). Use the

log11 ϩ r2

TABLE feature of a graphing calculator to help

estimate what interest rate is needed for an

investment to triple in nine years.

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College Algebra Graphs & Models—

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EXTENDING THE CONCEPT

107. Many texts and reference books give estimates of

the noise level (in decibels dB) of common sounds.

Through reading and research, try to locate or

approximate where the following sounds would fall

along this scale. In addition, determine at what point

pain or ear damage begins to occur.

a. threshold of audibility b. lawn mower

c. whisper

d. loud rock concert

e. lively party

f. jet engine

5–38

CHAPTER 5 Exponential and Logarithmic Functions

108. Determine the value of x that makes the equation

true: log3 3log3 1log3x2 4 ϭ 0.

109. Find the value of each expression without using a

calculator.

1

a. log6416

b. log4927

c. log0.2532

8

110. Suppose you and I represent two different numbers.

Is the following cryptogram true or false? The log

of me base me is one and the log of you base you is

one, but the log of you base me is equal to the log

of me base you turned upside down.

MAINTAINING YOUR SKILLS

3

111. (2.2) Graph g1x2 ϭ 2x ϩ 2 Ϫ 1 by shifting the

parent function. Then state the domain and range

of g.

112. (R.4) Factor the following expressions:

a. x3 Ϫ 8

b. a2 Ϫ 49

c. n2 Ϫ 10n ϩ 25

d. 2b2 Ϫ 7b ϩ 6

113. (4.2/4.6) For the graph

shown, write the solution

set for f 1x2 6 0. Then write

the equation of the graph in

factored form and in

polynomial form.

5.4

y

120

100

80

60

40

20

Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2

Ϫ20

2 4 6 8 10 x

Ϫ40

Ϫ60

Ϫ80

114. (2.1) A function f (x) is defined by the ordered pairs

shown in the table. Is the function (a) linear?

(b) increasing? Justify your answers.

x

y

Ϫ10

0

Ϫ9

Ϫ2

Ϫ8

Ϫ8

Ϫ6

Ϫ18

Ϫ5

Ϫ50

Ϫ4

Ϫ72

Properties of Logarithms

LEARNING OBJECTIVES

In Section 5.4 you will see

how we can:

A. Solve logarithmic

equations using the

fundamental properties

of logarithms

B. Apply the product,

quotient, and power

properties of logarithms

C. Apply the change-of-base

formula

D. Solve applications using

properties of logarithms

Logarithmic and exponential expressions have several fundamental properties that

enable us to solve some basic equations. In this section, we’ll learn how to use these

relationships effectively, and introduce additional properties that enable us to simplify

more complex equations before relying on the same fundamental properties to complete the solution.

A. Solving Equations Using the Fundamental

Properties of Logarithms

In Section 5.3, we converted expressions from exponential form to logarithmic form

using the basic definition: x ϭ by 3 y ϭ logb x. This relationship reveals the following fundamental properties:

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Section 5.4 Properties of Logarithms

517

Fundamental Properties of Logarithms

For any base b 7 0, b 1,

I. logb b ϭ 1, since b1 ϭ b

II. logb 1 ϭ 0, since b0 ϭ 1

III. logb bx ϭ x, since bx ϭ bx

IV. blogb x ϭ x 1x 7 02 , since logb x ϭ logb x

To see the verification of Property IV more clearly, again note that for

y ϭ logb x, b y ϭ x is the exponential form, and substituting logb x for y yields blogb x ϭ x.

Also note that Properties III and IV demonstrate that y ϭ logb x and y ϭ bx are inverse

functions. In common language, “a base-b logarithm undoes a base-b exponential,”

and “a base-b exponential undoes a base-b logarithm.” For f 1x2 ϭ log b x and

f Ϫ1 1x2 ϭ bx, using a composition verifies the inverse relationship just as in Example 5

from Section 5.1:

1 f ‫ ؠ‬f Ϫ1 2 1x2 ϭ f 3 f Ϫ1 1x2 4

1 f Ϫ1 ‫ ؠ‬f 21x2 ϭ f Ϫ1 3 f 1x2 4

ϭ logb bx

ϭx

ϭ blogb x

ϭx

These properties can be used to solve basic equations involving logarithms and exponentials. From the uniqueness property for exponents (page 496), note that if logb x ϭ k

, then blogb x ϭ bk, and we say that we have exponentiated both sides.

EXAMPLE 1

Solving Basic Logarithmic Equations

Solve each equation by applying fundamental properties. Answer in exact form and

approximate form using a calculator (round to 1000ths).

a. ln x ϭ 2

b. Ϫ0.52 ϭ log x

Solution

a. ln x ϭ 2

eln x ϭ e2

x ϭ e2

Ϸ 7.389

given

exponentiate both sides

Property IV, exact form

approximate form

b. Ϫ0.52 ϭ log x

10Ϫ0.52 ϭ 10log x

10Ϫ0.52 ϭ x

0.302 Ϸ x

given

exponentiate both sides

Property IV, exact form

approximate form

Now try Exercises 7 through 10 ᮣ

Note that checking the exact solutions by

substitution is a direct application of Property III

(Figure 5.38).

Also, we observe that exponentiating both sides

of the equation produces the same result as simply

writing the original equation in exponential form, and

the process can be viewed in terms of either

approach.

EXAMPLE 2

Figure 5.38

Solving Basic Exponential Equations

Solve each equation by applying fundamental properties. Answer in exact form and

approximate form using a calculator (round to 1000ths).

a. ex ϭ 167

b. 10 x ϭ 8.223

Solution

a.

ex ϭ 167

ln ex ϭ ln 167

x ϭ ln 167

x Ϸ 5.118

given

use natural log

Property III, exact form

approximate form

b.

10x ϭ 8.223

log 10x ϭ log 8.223

x ϭ log 8.223

x Ϸ 0.915

given

use common log

Property III, exact form

approximate form

Now try Exercises 11 through 14 ᮣ

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