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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ᮣ
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 ᮣ