3-3 Properties of Logarithms
Express each logarithm in terms of ln 2 and ln
5.
1. ln
ANSWER:
2 ln 5 – ln 2
5. ln
SOLUTION:
SOLUTION:
ANSWER:
2 ln 2 – ln 5
ANSWER:
ln 2 – ln 5
2. ln 200
SOLUTION:
6. ln
SOLUTION:
ANSWER:
3 ln 2 + 2 ln 5
3. ln 80
SOLUTION:
ANSWER:
ln 2 − ln 5
7. ln 2000
SOLUTION:
ANSWER:
4 ln 2 + ln 5
4. ln 12.5
SOLUTION:
ANSWER:
4 ln 2 + 3 ln 5
8. ln 1.6
SOLUTION:
ANSWER:
2 ln 5 – ln 2
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ln
ANSWER:
3 ln 2 – ln 5
SOLUTION:
Page 1
ANSWER: of Logarithms
3-3 Properties
4 ln 2 + 3 ln 5
8. ln 1.6
ANSWER:
2 ln 7 – 4 ln 3
11. ln
SOLUTION:
SOLUTION:
ANSWER:
ln 7 – 2 ln 3
ANSWER:
3 ln 2 – ln 5
Express each logarithm in terms of ln 3 and ln
7.
9. ln 63
12. ln 147
SOLUTION:
SOLUTION:
ANSWER:
ln 3 + 2 ln 7
13. ln 1323
ANSWER:
2 ln 3 + ln 7
SOLUTION:
10. ln
SOLUTION:
ANSWER:
3 ln 3 + 2 ln 7
14. ln
ANSWER:
SOLUTION:
2 ln 7 – 4 ln 3
11. ln
SOLUTION:
ANSWER:
3 ln 7 − 6 ln 3
15. ln
ANSWER:
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ln 7 – 2 ln 3
SOLUTION:
Page 2
c. If the concentration of hydrogen ions in a sample
–9
ANSWER: of Logarithms
3-3 Properties
3 ln 7 − 6 ln 3
of water is 1 × 10 moles per liter, what is the
concentration of hydroxide ions?
SOLUTION:
a.
15. ln
SOLUTION:
b.
ANSWER:
c.
4 ln 7 – 4 ln 3
16. ln 1701
SOLUTION:
1 × 10-5 moles per liter
ANSWER:
ANSWER:
–
+
a. log Kw = log [H ] + log [OH ]
5 ln 3 + ln 7
+
17. CHEMISTRY The ionization constant of water
Kw is the product of the concentrations of hydrogen
–
+
(H ) and hydroxide (OH ) ions.
–
b. –14 = log [H ] + log [OH ]
c. 1 × 10–5 moles per liter
18. TORNADOES The distance d in miles that a
tornado travels is
, where w is the wind
speed in miles per hour of the tornado.
a. Express w in terms of log d.
b. If a tornado travels 100 miles, estimate the wind
speed.
The formula for the ionization constant of water is
+
–
Kw = [H ][OH ], where the brackets denote
concentration in moles per liter.
+
a. Express log Kw in terms of log [H ] and log
SOLUTION:
a.
–
[OH ].
b. The value of the constant Kw is 1 × 10–14.
Simplify your equation from part a to reflect the
numerical value of Kw.
c. If the concentration of hydrogen ions in a sample
–9
b.
of water is 1 × 10 moles per liter, what is the
concentration of hydroxide ions?
SOLUTION:
a.
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Page 3
ANSWER:
b.
a. w = 93 log d + 65
ANSWER:
–
+
ANSWER:
a. log Kw = log [H ] + log [OH ]
+
3-3
–
b. –14 = log [H ] + log [OH ]
Properties
of Logarithms
c. 1 × 10–5 moles per liter
18. TORNADOES The distance d in miles that a
20. 8 ln e2 – ln e12
tornado travels is
, where w is the wind
speed in miles per hour of the tornado.
a. Express w in terms of log d.
b. If a tornado travels 100 miles, estimate the wind
speed.
SOLUTION:
SOLUTION:
ANSWER:
a.
4
21. 9 ln e3 + 4 ln e5
SOLUTION:
ANSWER:
b.
47
22.
SOLUTION:
ANSWER:
a. w = 93 log d + 65
b. 251 mph
ANSWER:
Evaluate each logarithm.
1
19.
SOLUTION:
23. 2 log3
SOLUTION:
ANSWER:
ANSWER:
3
24.
SOLUTION:
2
20. 8 ln e – ln e
12
SOLUTION:
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Page 4
ANSWER: of Logarithms
3-3 Properties
ANSWER:
3
24.
75
27.
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
1
25. 4 log2
28. 36 ln e0.5 – 4 ln e5
SOLUTION:
SOLUTION:
ANSWER:
–2
ANSWER:
6
Expand each expression.
29. log9 6x3y 5z
SOLUTION:
26. 50 log5
SOLUTION:
ANSWER:
log9 6 + 3 log9 x + 5 log9 y + log9 z
30.
ANSWER:
SOLUTION:
75
27.
SOLUTION:
ANSWER:
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ANSWER:
31.
Page 5
SOLUTION:
ANSWER:
ANSWER:
3-3 Properties of Logarithms
2 log7 h + 11 log7 j – 5 log7 k
35. log4 10t2uv−3
31.
SOLUTION:
SOLUTION:
ANSWER:
log4 10 + 2 log4 t + log4 u – 3log4 v
36. log5 a 6b −3c4
ANSWER:
SOLUTION:
32.
ANSWER:
6 log5 a – 3 log5 b + 4 log5 c
SOLUTION:
37.
SOLUTION:
ANSWER:
ANSWER:
33. log11 ab−4c12d 7
SOLUTION:
38.
ANSWER:
SOLUTION:
log11 a – 4 log11 b + 12 log11 c + 7 log11 d
34. log7 h 2j 11k −5
SOLUTION:
ANSWER:
ANSWER:
2 log7 h + 11 log7 j – 5 log7 k
35. log4 10t2uv−3
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Condense each expression.
Page 6
39.
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
3-3 Properties of Logarithms
Condense each expression.
42.
39.
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
43. 2 log8 (9x) – log8 (2x – 5)
SOLUTION:
40.
SOLUTION:
ANSWER:
ANSWER:
44. ln 13 + 7 ln a – 11 ln b + ln c
SOLUTION:
41. 7 log3 a + log3 b – 2 log3 (8c)
SOLUTION:
ANSWER:
7 –11
ln 13a b
ANSWER:
c
45. 2 log6 (5a) + log6 b + 7 log6 c
SOLUTION:
42.
SOLUTION:
ANSWER:
2
7
log6 25a bc
46. log2 x – log2 y – 3 log2 z
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ANSWER:
SOLUTION:
Page 7
ANSWER:
ANSWER:
2 7
3-3 Properties
log 25a bc of Logarithms
6
46. log2 x – log2 y – 3 log2 z
SOLUTION:
Evaluate each logarithm.
49. log6 14
SOLUTION:
ANSWER:
ANSWER:
1.473
50. log3 10
SOLUTION:
47.
SOLUTION:
ANSWER:
2.096
ANSWER:
51. log7 5
SOLUTION:
48.
ANSWER:
SOLUTION:
0.827
52. log128 2
SOLUTION:
ANSWER:
ANSWER:
0.143
Evaluate each logarithm.
49. log6 14
53. log12 145
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
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2.003
1.473
54. log22 400
Page 8
SOLUTION:
no real solution
ANSWER: of Logarithms
3-3 Properties
0.143
53. log12 145
ANSWER:
no real solution
58. log13,000 13
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
2.003
0.271
54. log22 400
SOLUTION:
ANSWER:
1.938
55. log100 101
SOLUTION:
59. COMPUTERS Computer programs are written
in sets of instructions called algorithms. To
execute a task in a computer program, the
algorithm coding in the program must be analyzed.
The running time in seconds R that it takes to
analyze an algorithm of n steps can be modeled by
R = log2 n.
a. Determine the running time to analyze an
algorithm of 240 steps.
b. To the nearest step, how many steps are in an
algorithm with a running time of 8.45 seconds?
SOLUTION:
a.
ANSWER:
1.002
b.
56.
SOLUTION:
ANSWER:
ANSWER:
1.585
57. log−2 8
a. 7.9 seconds
b. 350 steps
60. TRUCKING Bill’s Trucking Service purchased a
new delivery truck for $56,000. Suppose t = log(1 –
represents the time t in years that has passed
SOLUTION:
r)
no real solution
since the purchase given its initial price P, present
value V, and annual rate of depreciation r.
a. If the truck's present value is $40,000 and it has
depreciated at a rate of 15% per year, how much
time has passed since its purchase to the nearest
year?
Page 9
b. If the truck's present value is $34,000 and it has
depreciated at a rate of 10% per year, how much
ANSWER:
no real solution
58. logManual
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13,000 -13
SOLUTION:
about 5 years
ANSWER:
a. 7.9 secondsof Logarithms
3-3 Properties
b. 350 steps
60. TRUCKING Bill’s Trucking Service purchased a
new delivery truck for $56,000. Suppose t = log(1 –
r)
represents the time t in years that has passed
since the purchase given its initial price P, present
value V, and annual rate of depreciation r.
a. If the truck's present value is $40,000 and it has
depreciated at a rate of 15% per year, how much
time has passed since its purchase to the nearest
year?
b. If the truck's present value is $34,000 and it has
depreciated at a rate of 10% per year, how much
time has passed since its purchase to the nearest
year?
ANSWER:
a. 2 yr
b. 5 yr
Estimate each logarithm to the nearest whole
number.
61. log4 5
SOLUTION:
ANSWER:
1
62. log2 13
SOLUTION:
SOLUTION:
a.
ANSWER:
4
63. log3 10
SOLUTION:
about 2 years
b.
ANSWER:
2
64. log7 400
SOLUTION:
ANSWER:
about 5 years
ANSWER:
a. 2 yr
b. 5 yr
3
65. log5
SOLUTION:
Estimate each logarithm to the nearest whole
number.
61. log4 5
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SOLUTION:
Page 10
ANSWER:
ANSWER: of Logarithms
3-3 Properties
3
1
65. log5
68. log4
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
–3
–4
Expand each expression.
66. log12 177
SOLUTION:
69.
SOLUTION:
ANSWER:
2
67.
ANSWER:
SOLUTION:
ln x +
ANSWER:
ln (x + 3)
70.
1
SOLUTION:
68. log4
SOLUTION:
ANSWER:
ANSWER:
71.
–4
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Expand each expression.
69.
SOLUTION:
Page 11
ANSWER:
ANSWER:
3-3 Properties of Logarithms
71.
log8 x +
log8 y +
log8 (z – 1)
74.
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
75. EARTHQUAKES The Richter scale measures
the intensity of an earthquake. The magnitude M of
the seismic energy in joules E released by an
earthquake can be calculated by
.
72.
SOLUTION:
ANSWER:
2 ln 3 + 2 ln x + ln y + 3 ln z – 4 ln (y – 5)
73.
a. Use the properties of logarithms to expand the
equation.
b. What magnitude would an earthquake releasing
11
7.94 × 10 joules have?
c. The 2007 Alum Rock earthquake in California
12
released 4.47 × 10 joules of energy. The 1964
Anchorage earthquake in Alaska measured a
SOLUTION:
18
ANSWER:
log8 x +
log8 y +
log8 (z – 1)
magnitude of 1.58 × 10 joules of energy. How
many times as great was the magnitude of the
Anchorage earthquake as the magnitude of the
Alum Rock earthquake?
d. Generally, earthquakes cannot be felt until they
reach a magnitude of 3 on the Richter scale. How
many joules of energy does an earthquake of this
magnitude release?
SOLUTION:
74.
a.
SOLUTION:
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Page 12
reach a magnitude of 3 on the Richter scale. How
many joules of energy does an earthquake of this
magnitude release?
a.
b. 5
c. 1.67
d. 7.94 × 108 joules
3-3 Properties
SOLUTION: of Logarithms
a.
Condense each expression.
76.
ln x +
ln y +
ln z
SOLUTION:
b.
ANSWER:
c.
77.
SOLUTION:
ANSWER:
a.
b. 5
c. 1.67
d. 7.94 × 108 joules
Condense each expression.
76.
ln x +
ln y +
ln z
SOLUTION:
ANSWER:
log2
or log2
78.
SOLUTION:
ANSWER:
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Page 13
ANSWER:
ANSWER:
log2
or log2
3-3 Properties
of Logarithms
79.
78.
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
80.
79.
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
or log4
log4
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81.
ln x +
ln (y + 8) – 3 ln y – ln (10 – x)
80.
SOLUTION:
Page 14
ANSWER:
ANSWER:
3-3
log4
Properties
81.
ln x +
of
or log4
Logarithms
ln (y + 8) – 3 ln y – ln (10 – x)
2 ln 2; 1.38
83. ln 48
SOLUTION:
SOLUTION:
ANSWER:
4 ln 2 + ln 3; 3.86
84. ln 162
SOLUTION:
ANSWER:
Use the properties of logarithms to rewrite
each logarithm below in the form a ln 2 + b ln
3, where a and b are constants. Then
approximate the value of each logarithm given
that ln 2 ≈ 0.69 and ln 3 ≈ 1.10.
82. ln 4
ANSWER:
ln 2 + 4 ln 3; 5.09
85. ln 216
SOLUTION:
SOLUTION:
ANSWER:
2 ln 2; 1.38
ANSWER:
3 ln 2 + 3 ln 3; 5.37
83. ln 48
SOLUTION:
86. ln
SOLUTION:
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ANSWER:
4 ln 2 + ln 3; 3.86
ANSWER:
ln 3 – ln 2; 0.41
Page 15
ANSWER: of Logarithms
3-3 Properties
3 ln 2 + 3 ln 3; 5.37
86. ln
SOLUTION:
ANSWER:
2 ln 2 – 3 ln 3; –1.92
89. ln
SOLUTION:
ANSWER:
ln 3 – ln 2; 0.41
ANSWER:
87. ln
SOLUTION:
5 ln 2 – 2 ln 3; 1.25
Determine the graph that corresponds to each
equation.
ANSWER:
2 ln 2 − 2 ln 3; –0.82
88. ln
SOLUTION:
90. f (x) = ln x + ln (x + 3)
SOLUTION:
ANSWER:
2 ln 2 – 3 ln 3; –1.92
Make a table of values.
x
0
1
2
3
4
89. ln
SOLUTION:
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f(x)
undefined
1.39
2.30
2.89
3.33
This table resembles the graphs for a, c, and d.
However, the origin is a point on the graph of d. f
(x) is undefined for x = 1 or x = 2 in graph c. The
correct choice is a.
Page 16
ANSWER:
a
correct choice is a.
ANSWER: of Logarithms
3-3 Properties
5 ln 2 – 2 ln 3; 1.25
Determine the graph that corresponds to each
equation.
ANSWER:
a
91. f (x) = ln x – ln (x + 5)
SOLUTION:
Make a table of values.
x
0
1
2
3
4
f(x)
undefined
−1.79
−1.25
−0.98
−0.69
This table resembles graph b.
ANSWER:
b
90. f (x) = ln x + ln (x + 3)
92. f (x) = 2 ln (x + 1)
SOLUTION:
SOLUTION:
Make a table of values.
x
0
1
2
3
4
Make a table of values.
x
0
1
2
3
4
f(x)
undefined
1.39
2.30
2.89
3.33
This table resembles the graphs for a, c, and d.
However, the origin is a point on the graph of d. f
(x) is undefined for x = 1 or x = 2 in graph c. The
correct choice is a.
ANSWER:
a
f(x)
0
0.48
1.21
1.92
2.59
f(x) is undefined for x ≤ 0 for graph a.
This table resembles graph d.
ANSWER:
d
93. f (x) = 0.5 ln (x – 2)
SOLUTION:
91. f (x) = ln x – ln (x + 5)
Make a table of values.
x
2
3
4
5
6
SOLUTION:
Make a table of values.
x
f(x)
0
undefined
1
−1.79
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2
−1.25
3
−0.98
4
−0.69
f(x)
undefined
0
0.35
0.55
0.69
f(x) is undefined for x ≤ 2.
This table resembles graph c.
ANSWER:
c
Page 17
f(x) is undefined for x ≤ 0 for graph a.
This table resembles graph d.
ANSWER: of Logarithms
3-3 Properties
d
2
undefined
This table resembles graph f.
ANSWER:
f
93. f (x) = 0.5 ln (x – 2)
95. f (x) = ln 2x – 4 ln x
SOLUTION:
SOLUTION:
Make a table of values.
x
2
3
4
5
6
f(x)
undefined
0
0.35
0.55
0.69
Make a table of values.
x
0
1
2
3
4
f(x) is undefined for x ≤ 2.
This table resembles graph c.
ANSWER:
c
94. f (x) = ln (2 – x) + 6
SOLUTION:
Make a table of values.
x
−2
−1
0
1
2
f(x)
7.4
7.1
6.7
6
undefined
This table resembles graph f.
f(x)
undefined
0.69
−1.39
−2.60
−3.47
This table resembles graph e .
ANSWER:
e
Write each set of logarithmic expressions in
increasing order.
96. log3
, log3
+ log3 4, log3 12 – 2 log3 4
SOLUTION:
ANSWER:
f
95. f (x) = ln 2x – 4 ln x
SOLUTION:
Make a table of values.
x
0
1
2
3
4
f(x)
undefined
0.69
−1.39
−2.60
−3.47
This table resembles graph e .
ANSWER:
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e
Write each set of logarithmic expressions in
ANSWER:
log3 12 – 2 log3 4, log3
97. log5 55, log5
SOLUTION:
, 3 log5
, log3
+ log3 4
Page 18
ANSWER:
log3 12 – 2 log3 4, log3
, log3
3-3 Properties
of Logarithms
97. log5 55, log5
+ log3 4
ANSWER:
264 hours
Write an equation for each graph.
, 3 log5
SOLUTION:
99.
SOLUTION:
Points (1, 0) and (10, 1) are located on the graph.
Point (1, 0) indicates that there are no translations
from the parent graph.
ANSWER:
log5
, log5 55, 3 log5
Use (10, 1) to identify the base.
98. BIOLOGY The generation time for bacteria is the
time that it takes for the population to double. The
generation time G can be found using G =
, where t is the time period, b is the
number of bacteria at the beginning of the
experiment, and f is the number of bacteria at the
end of the experiment. The generation time for
mycobacterium tuberculosis is 16 hours. How long
will it take four of these bacteria to multiply into
1024 bacteria?
ANSWER:
f(x) = log10 x
SOLUTION:
100.
SOLUTION:
Points (1, 0) and (8, −3) are located on the graph.
ANSWER:
Point (1, 0) indicates that there are no translations
from the parent graph.
264 hours
Write an equation for each graph.
99.
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SOLUTION:
Points (1, 0) and (10, 1) are located on the graph.
Use (8, −3) to identify the base.
Page 19
ANSWER:
ANSWER:
f(x) = log10 x of Logarithms
3-3 Properties
100.
101.
SOLUTION:
SOLUTION:
Points (1, 0) and (8, −3) are located on the graph.
Points (1, 0) and (8, 3) are located on the graph.
Point (1, 0) indicates that there are no translations
from the parent graph.
Point (1, 0) indicates that there are no translations
from the parent graph.
Use (8, −3) to identify the base.
Use (8, 3) to identify the base.
ANSWER:
h(x) = log2 x
ANSWER:
102.
SOLUTION:
Points (1, 0) and (1500, 1) are located on the graph.
Point (1, 0) indicates that there are no translations
from the parent graph.
101.
SOLUTION:
Use (1500, 1) to identify the base.
Points (1, 0) and (8, 3) are located on the graph.
Point (1, 0) indicates that there are no translations
from the parent graph.
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Use (8, 3) to identify the base.
Page 20
ANSWER:
k(x) = log1500 x
The Ka of a substance can be calculated by Ka =
. If a substance has a pKa = 25,
ANSWER:
3-3 Properties
of Logarithms
h(x) = log x
what is its Ka?
2
d. Aldehydes are a common functional group in
organic molecules. Aldehydes have a pKa around
17. To what Ka does this correspond?
SOLUTION:
a.
102.
SOLUTION:
Points (1, 0) and (1500, 1) are located on the graph.
b.
Point (1, 0) indicates that there are no translations
from the parent graph.
Use (1500, 1) to identify the base.
c.
ANSWER:
k(x) = log1500 x
103. CHEMISTRY pKa is the logarithmic acid
dissociation constant for the acid HF, which is
–
+
composed of ions H and F . The pKa can be
calculated by pKa = –log
+
, where
+
d.
−
[H ] is the concentration of H ions, [F ] is the
–
concentration of F ions, and [HF] is the
concentration of the acid solution. All of the
concentrations are measured in moles per liter.
a. Use the properties of logs to expand the equation
for pKa.
b. What is the pKa of a reaction in which [H+] =
–
0.01 moles per liter, [F ] = 0.01 moles per liter, and
[HF] = 2 moles per liter?
c. The Ka of a substance can be calculated by Ka =
. If a substance has a pKa = 25,
ANSWER:
what is its Ka?
d. Aldehydes are a common functional group in
organic molecules. Aldehydes have a pKa around
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+
–
a. pKa = –(log [H ] + log [F ] – log [HF])
b. pKa = 4.30
17. To what Ka does this correspond?
c. Ka = 1 × 10–25
SOLUTION:
d. Ka = 1 × 10–17
Page 21
ANSWER:
ANSWER: of Logarithms
3-3 Properties
+
c. Ka = 1 × 10–25
d. Ka = 1 × 10
8
–
a. pKa = –(log [H ] + log [F ] – log [HF])
b. pKa = 4.30
107.
SOLUTION:
–17
Evaluate each expression.
ANSWER:
104.
2
SOLUTION:
Simplify each expression.
108. (log3 6)(log6 13)
SOLUTION:
ANSWER:
6
105.
ANSWER:
log3 13
SOLUTION:
109. (log2 7)(log5 2)
SOLUTION:
ANSWER:
4
106.
SOLUTION:
ANSWER:
log5 7
110. (log4 9) ÷ (log4 2)
SOLUTION:
ANSWER:
8
107.
SOLUTION:
ANSWER:
ANSWER:
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eSolutions
Simplify each expression.
108. (log 6)(log 13)
log2 9
111.
Page 22
3-3 Properties of Logarithms
intensity of the light from the screen that reaches
the viewer and K is the constant of proportionality.
a. The intensity of the light perceived by a
moviegoer who sits at a distance d from the screen
is given by
, where k is a constant of
proportionality. Show that f = K(log k − 2 log d).
110. (log4 9) ÷ (log4 2)
SOLUTION:
b. Suppose you notice the flicker from a movie
projection and move to double your distance from
the screen. In terms of K, how does this move
affect the value of f ? Explain your reasoning.
SOLUTION:
ANSWER:
a.
log2 9
111. (log5 12) ÷ (log8 12)
SOLUTION:
b. Let f 1 be the minimum frequency at which the
flicker first disappears when you were in your
original seat at distance d from the screen. Let f 2
be the minimum frequency when your distance
from the screen is 2d. Then
ANSWER:
log5 8
112. MOVIES Traditional movies are a sequence of
still pictures which, if shown fast enough, give the
viewer the impression of motion. If the frequency
of the stills shown is too small, the moviegoer
notices a flicker between each picture. Suppose the
minimum frequency f at which the flicker first
disappears is given by f = K log I, where I the
intensity of the light from the screen that reaches
the viewer and K is the constant of proportionality.
a. The intensity of the light perceived by a
moviegoer who sits at a distance d from the screen
is given by
, where k is a constant of
So the effect is f 2 − f 1.
ANSWER:
a.
proportionality. Show that f = K(log k − 2 log d).
eSolutions Manual - Powered by Cognero
b. Suppose you notice the flicker from a movie
projection and move to double your distance from
the screen. In terms of K, how does this move
b. Let f 1 be the minimum frequency at whichPage
the 23
flicker first disappears when you were in your
original seat at distance d from the screen. Let f 2
The minimum frequency at which the flicker first
disappears is reduced by 0.602K.
3-3 Properties of Logarithms
PROOF Investigate graphically and then
prove each of the following properties of
logarithms.
113. Quotient Property
b. Let f 1 be the minimum frequency at which the
flicker first disappears when you were in your
original seat at distance d from the screen. Let f 2
be the minimum frequency when your distance
from the screen is 2d. Then
SOLUTION:
The quotient property is
.
Let g(x) = ln (x + 2) and h(x) = ln (x + 4). The
graph of f (x) represents the difference between the
graphs of h(x) and g(x). Then f (x) =
.
So the effect is f 2 – f 1.
Let x = logb m Let y = logb n. Then by the
The minimum frequency at which the flicker first
disappears is reduced by 0.602K.
x
y
definition of logarithms, b = m and b = n.
PROOF Investigate graphically and then
prove each of the following properties of
logarithms.
113. Quotient Property
SOLUTION:
The quotient property is
.
ANSWER:
Let g(x) = ln (x + 2) and h(x) = ln (x + 4). The
graph of f (x) represents the difference between the
graphs of h(x) and g(x). Then f (x) =
.
The graph of f (x) represents the difference
between the graphs of h(x) and g(x).
x
y
Let x = logb m Let y = logb n. Then b = m and b
= n.
Let x = logb m Let y = logb n. Then by the
x
y
definition of logarithms, b = m and b = n.
114. Power Property
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ANSWER:
SOLUTION:
Page 24
Let f (x) = ln x.. Then identify g(x) such that all of
the function values of g(x) are twice the function
3-3 Properties of Logarithms
114. Power Property
115. PROOF Prove that logb x =
.
SOLUTION:
Let f (x) = ln x.. Then identify g(x) such that all of
the function values of g(x) are twice the function
values of f (x) for each x-value in the domain. .
Then
SOLUTION:
Prove the change of base formula.
.
ANSWER:
116. REASONING How can the graph of g(x) = log4
x be obtained using a transformation of the graph of
f (x) = ln x?
ANSWER:
SOLUTION:
First, express f (x) = ln x and g(x) = log4 x in similar
terms.
All of the function values of g(x) are twice the
function values of f (x) for each x-value in the
domain.
In order to transform the graph of f (x) to the graph
of g(x), we need to multiply
115. PROOF Prove that logb x =
.
u that will yield
by a value
.
SOLUTION:
Prove the change of base formula.
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Page 25
ANSWER:
Sample answer: Expand f (x) = ln x by a factor of
3-3 Properties of Logarithms
.
116. REASONING How can the graph of g(x) = log4
117. CHALLENGE If
, for what values of x can
x be obtained using a transformation of the graph of
f (x) = ln x?
ln x not be simplified?
SOLUTION:
ln x cannot be simplified for values of x that are
prime numbers. The rest of the numbers can be
factored.
First, express f (x) = ln x and g(x) = log4 x in similar
terms.
SOLUTION:
For example,
ln 4 = 2 ln 2 because 4 is 2 × 2.
ln 15 = ln 3 + ln 5 because 15 is 3 × 5.
ln 7 cannot be simplified because 7 has no factors
besides 1 and 7. ln 7 + ln 1 = ln 7 + 0 = ln 7.
In order to transform the graph of f (x) to the graph
of g(x), we need to multiply
u that will yield
by a value
.
ANSWER:
ln x cannot be simplified for values of x that are
prime numbers.
118. ERROR ANALYSIS Omar and Nate expanded
using the properties of logarithms. Is
either of them correct? Explain.
4 log2 x + 4 log2 y – 4 log2 z
Omar:
Nate:
2 log4 x + 2 log4 y – 2 log4 z
SOLUTION:
Omar; Nate incorrectly transposed the exponent
and the base.
ANSWER:
Omar; Nate incorrectly transposed the exponent
and the base.
ANSWER:
Sample answer: Expand f (x) = ln x by a factor of
119. PROOF Use logarithmic properties to prove
.
117. CHALLENGE If
, for what values of x can
SOLUTION:
ln x not be simplified?
SOLUTION:
eSolutions Manual - Powered by Cognero
ln x cannot be simplified for values of x that are
prime numbers. The rest of the numbers can be
factored.
Page 26
and the base.
ANSWER:
Omar; Nate incorrectly
transposed the exponent
3-3 Properties
of Logarithms
and the base.
119. PROOF Use logarithmic properties to prove
SOLUTION:
120. Writing in Math The graph of g(x) = logb x is
actually a transformation of f (x) = log x. Use the
Change of Base Formula to find the transformation
that relates these two graphs. Then explain the
effect that different values of b have on the
common logarithm graph.
SOLUTION:
The change of base formula is
.
Rewrite logb x in terms of log x.
logb x =
=
log x. Thus, a base b
logarithm is a constant multiple of its corresponding
common logarithm.
ANSWER:
When b > 1, the graph of f is expanded or
compressed vertically. For example, if b = 2, the
graph will be expanded, but when b = 25, the graph
will be compressed. When b < 1, in addition to
being expanded or compressed vertically, the graph
is reflected in the x-axis.
ANSWER:
Sample answer: logb x =
=
log x. Thus,
a base b logarithm is a constant multiple of its
corresponding common logarithm. When b > 1, the
graph of f is expanded or compressed vertically.
For example, if b = 2, the graph will be expanded,
but when b = 25, the graph will be compressed.
When b < 1, in addition to being expanded or
compressed vertically, the graph is reflected in the
x-axis.
120. Writing in Math The graph of g(x) = logb x is
actually a transformation of f (x) = log x. Use the
Change of Base Formula to find the transformation
that relates these two graphs. Then explain the
effect
that- Powered
different
eSolutions
Manual
by values
Cognero of b have on the
common logarithm graph.
SOLUTION:
Sketch and analyze each function. Describe its
domain, range, intercepts, asymptotes, end
behavior, and where the function is increasing
or decreasing.
121. f (x) = log6 x
Page 27
SOLUTION:
Evaluate the function for several x-values in its
D = (0, ); R = (–
asymptote: y-axis;
compressed vertically, the graph is reflected in the
x-axis.
,
); x-intercept: 1;
increasing on (0,
Sketch and analyze
each function. Describe its
3-3 Properties
of Logarithms
domain, range, intercepts, asymptotes, end
behavior, and where the function is increasing
or decreasing.
121. f (x) = log6 x
)
122.
SOLUTION:
SOLUTION:
Evaluate the function for several x-values in its
domain.
Evaluate the function for several x-values in its
domain.
x
0
1
2
3
4
5
6
y
undefined
0
0.39
0.61
0.77
0.90
1
x
0
1
2
3
4
5
6
y
undefined
0
0.63
−1
−1.26
−1.47
−1.63
Then use a smooth curve to connect each of these
ordered pairs.
Then use a smooth curve to connect each of these
ordered pairs.
List the domain, range, intercepts, asymptotes, end
behavior, and where the function is increasing or
decreasing.
List the domain, range, intercepts, asymptotes, end
behavior, and where the function is increasing or
decreasing.
D = (0, ); R = (–
asymptote: y-axis;
D = (0, ); R = (–
asymptote: y-axis;
,
); x-intercept: 1;
,
decreasing on (0,
increasing on (0,
)
)
ANSWER:
ANSWER:
D = (0, ); R = (–
asymptote: y-axis;
); x-intercept: 1;
,
); x-intercept: 1;
D = (0, ); R = (–
asymptote: y-axis;
122. Manual - Powered by Cognero
eSolutions
); x-intercept: 1;
decreasing on (0,
increasing on (0,
)
,
)
123. h(x) = log5 x − 2
Page 28
SOLUTION:
SOLUTION:
Evaluate the function for several x-values in its
D = (0, ); R = (–
asymptote: y-axis;
,
); x-intercept: 1;
3-3 Properties of Logarithms
decreasing on (0,
)
D = (0, ); R = (–
asymptote: y-axis;
123. h(x) = log5 x − 2
); x-intercept: 25;
increasing on (0,
SOLUTION:
Evaluate the function for several x-values in its
domain.
x
0
1
2
3
4
5
6
,
y
undefined
−2
−1.57
−1.32
−1.14
−1
−0.89
)
Use the graph of f (x) and g(x) to describe the
transformation that yields the graph of g(x).
Then sketch the graphs of f (x) and g(x).
124. f (x) = 2x; g(x) = −2x
SOLUTION:
x
This function is of the form f (x) = 2 . Rewrite g(x)
in terms of f (x).
Then use a smooth curve to connect each of these
ordered pairs.
g(x) = −f (x)
Therefore the graph of g(x) is the graph of f (x)
reflected in the x-axis.
List the domain, range, intercepts, asymptotes, end
behavior, and where the function is increasing or
decreasing.
To determine the intercept, set h(x) = 0.
ANSWER:
g(x) is the graph of f (x) reflected in the x-axis.
D = (0, ); R = (–
asymptote: y-axis;
,
); x-intercept: 25;
increasing on (0,
)
ANSWER:
125. f (x) = 5x; g(x) = 5x + 3
SOLUTION:
x
The function is of the form f (x) = 5 . Rewrite g(x)
in terms of f (x).
g(x) = f (x + 3)
D = (0,
); R = (–
,
); x-intercept: 25;
asymptote:
y-axis;by Cognero
eSolutions
Manual - Powered
Page 29
increasing on (0,
)
Therefore the graph of g(x) is the graph of f (x)
translated 3 units left.
3-3 Properties of Logarithms
125. f (x) = 5x; g(x) = 5x + 3
126.
SOLUTION:
x
The function is of the form f (x) = 5 . Rewrite g(x)
in terms of f (x).
g(x) = f (x + 3)
Therefore the graph of g(x) is the graph of f (x)
translated 3 units left.
SOLUTION:
This function is of the form f (x) =
(x) in terms of f (x).
. Rewrite g
g(x) = f (x) − 2
Therefore the graph of g(x) is the graph of f (x)
translated 2 units down.
ANSWER:
g(x) is the graph of f (x) translated 3 units left.
ANSWER:
g(x) is the graph of f (x) translated 2 units down.
126.
127. GEOMETRY The volume of a rectangular prism
SOLUTION:
This function is of the form f (x) =
(x) in terms of f (x).
with a square base is fixed at 120 cubic feet.
. Rewrite g
g(x) = f (x) − 2
Therefore the graph of g(x) is the graph of f (x)
translated 2 units down.
a. Write the surface area of the prism as a function
A(x) of the length of the side of the square x.
b. Graph the surface area function.
c. What happens to the surface area of the prism
as the length of the side of the square approaches
0?
SOLUTION:
a.
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ANSWER:
Page 30
3-3
b. Graph the surface area function.
c. What happens to the surface area of the prism
as the length of the side of the square approaches
0?
Properties
of Logarithms
SOLUTION:
a.
c. The surface area approaches infinity.
Divide using synthetic division.
128. (x2 – x + 4) ÷ (x – 2)
SOLUTION:
b.
ANSWER:
129. (x3 + x2 – 17x + 15) ÷ (x + 5)
SOLUTION:
c. As x approaches 0,
approaches infinity,
and as indicated in the graph, A(x) also approaches
infinity. The surface area approaches infinity.
ANSWER:
a. A(x) =
+ 2x
2
b.
2
x – 4x + 3
ANSWER:
2
x – 4x + 3
130. (x3 – x2 + 2) ÷ (x + 1)
c. The surface area approaches infinity.
SOLUTION:
Divide using synthetic division.
128. (x2 – x + 4) ÷ (x – 2)
SOLUTION:
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Page 31
2
x – 2x + 2
ANSWER:
2
x – 4x + 3
x – 2x + 2
ANSWER:
Show that f and g are inverse functions. Then
graph each function on the same graphing
calculator screen.
2
3-3 Properties
of Logarithms
2
x – 4x + 3
130. (x3 – x2 + 2) ÷ (x + 1)
131.
SOLUTION:
SOLUTION:
2
x – 2x + 2
ANSWER:
2
x – 2x + 2
Show that f and g are inverse functions. Then
graph each function on the same graphing
calculator screen.
131.
ANSWER:
SOLUTION:
ANSWER:
132. f (x) =
eSolutions Manual - Powered by Cognero
Page 32
g(x) =
–2
3-3 Properties of Logarithms
133. f (x) = (x – 3)3 + 4
132. f (x) =
g(x) =
–2
SOLUTION:
SOLUTION:
ANSWER:
ANSWER:
134. SCIENCE Specific heat is the amount of energy
133. f (x) = (x – 3)3 + 4
eSolutions Manual - Powered by Cognero
SOLUTION:
per unit of mass required to raise the temperature
of a substance by one degree Celsius. The table
lists the specific heat in joules per gram for certain
substances. The amount of energy transferred is
given by Q = cmT, where c is the specific heatPage
for33
a substance, m its mass, and T is the change in
temperature.
a. T =
b. f (x) =
3-3 Properties of Logarithms
c. D = {m | m > 0}
134. SCIENCE Specific heat is the amount of energy
per unit of mass required to raise the temperature
of a substance by one degree Celsius. The table
lists the specific heat in joules per gram for certain
substances. The amount of energy transferred is
given by Q = cmT, where c is the specific heat for
a substance, m its mass, and T is the change in
temperature.
135. SAT/ACT If b ≠ 0, let a
b=
. If x
y=
1, then which statement must be true?
A x =y
B x = –y
C x2 – y 2 = 0
D x > 0 and y > 0
E x = |y|
SOLUTION:
a. Find the function for the change in temperature.
b. What is the parent graph of this function?
c. What is the relevant domain of this function?
SOLUTION:
The correct choice is C.
ANSWER:
C
136. REVIEW Find the value of x for log2 (9x + 5) = 2
a.
2
+ log2 (x – 1).
F –0.4
G0
H1
J3
b. The function appears to be rational. f (x) =
c. D = {m | m > 0}. You cannot have negative
mass.
SOLUTION:
ANSWER:
a. T =
b. f (x) =
c. D = {m | m > 0}
135. SAT/ACT If b ≠ 0, let a
b=
. If x
1, then which statement must be true?
A x =y
B x = –y
C x2 – y 2 = 0
D x > 0 and y > 0
E x = |y|
SOLUTION:
eSolutions
Manual - Powered by Cognero
y=
2
Since x − 1 > 0, x > 1, so the correct choice is J.
ANSWER:
J
Page 34
137. To what is 2 log5 12 – log5 8 – 2 log5 3 equal?
A log5 2
The correct choice is C.
ANSWER: of Logarithms
3-3 Properties
C
The correct choice is A.
ANSWER:
A
136. REVIEW Find the value of x for log2 (9x + 5) = 2
2
+ log2 (x – 1).
F –0.4
G0
H1
J3
SOLUTION:
138. REVIEW The weight of a bar of soap decreases
by 2.5% each time it is used. If the bar of soap
weighs 95 grams when it is new, what is its weight
to the nearest gram after 15 uses?
F 58 g
G 59 g
H 65 g
J 93 g
SOLUTION:
If the soap decreases by 2.5% after each use, then
97.5% remains.
15
95 · 0.975
65
The correct choice is H.
ANSWER:
H
2
Since x − 1 > 0, x > 1, so the correct choice is J.
ANSWER:
J
137. To what is 2 log5 12 – log5 8 – 2 log5 3 equal?
A log5 2
B log5 3
C log5 0.5
D1
SOLUTION:
The correct choice is A.
ANSWER:
A
138. REVIEW The weight of a bar of soap decreases
by 2.5% each time it is used. If the bar of soap
weighs 95 grams when it is new, what is its weight
to the nearest gram after 15 uses?
F 58 g
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Page 35