Section 12.4 Exponential and Logarithmic Equations
37.
2
2
7410
ln 7 ln 410
x
x
38.
3
3
5137
ln 5 ln137
x
x
39.
1
25
xx
1
ln 2 ln 5
1ln2 ln5
ln 2 ln 2 ln 5
xx
xx
xx
40.
1
49
xx
1
ln 4 ln 9
ln 9 ln 4
xx
The solution set is ln 4 1.71
ln 9 ln 4
.
5
3
5
125
x
x
The solution set is {125}.
Chapter 12 Exponential and Logarithmic Functions
45.
9
1
log 2
x
46.
25
1
2
1
log 2
25
x
x
48.
3
log 3
10
1000
x
x
x
The solution set is {1000}.
51.
3
log 4 3x
14
27
1108109
27 27 27
x
x
12
49
198 97
49 49 49
x
x
97
Section 12.4 Exponential and Logarithmic Equations
54.
2
5
log 4 1 5
412
x
x
55.
ln 2
2
ln 2
7.39
x
x
ee
xe
The solution set is
27.39 .e
57.
3
3
ln 3
1
x
xe e
The solution set is
3
10.05 .ee
59.
5ln 2 20
ln 2 4
x
x
60.
6ln 2 30
ln 2 5
x
x
61.
1
ln 2
1
2
62ln 5
2ln 1
x
x
x
ee
1
ln 3
1
3
1
ln 3
0.72
x
x
ee
xe
ln 3 2
2
3
x
ee
xe
Chapter 12 Exponential and Logarithmic Functions
64. ln 4 1x
1
2
ln41
x
65.
55
5
1
log log 4 1 1
log 4 1 1
415
xx
xx
xx
equation. The solution set is 5.
4
66.
66
6
2
log 5 log 2
log 5 2
56
xx
xx
xx
67.
33
3
2
log 5 log 3 2
log 5 3 2
xx
xx
the logarithm of a negative number in the original
equation. The solution set is
6.
68.
22
log1log13
log 1 1 3
xx
xx
2
2
log 3
5
x
x
3
22
5
28
5
x
x
x
x
Section 12.4 Exponential and Logarithmic Equations
70.
44
log 2 log 1 1xx
4
2
log 1
1
x
x
71.
log 3 5 log 5 2xx
2
35
log 2
5
35
10
5
x
x
x
x
72.
log 2 1 log 2xx
2
21
log 2
21
10
x
x
x
x
73.
ln 1 ln 1xx
1
ln 1
x
74.
ln ln 2
2x
x
2
2
ln 2
2
x
x
x
ex
47
3
x
x
The solution set is
.
3
76.
22
log log 4
5
54
x
x
Chapter 12 Exponential and Logarithmic Functions
78. log(5 1) log(2 3) log 2xx
log(5 1) log(4 6)
xx
79. log(3 3) log( 1) log 4xx
log(3 3) log(4 4)
3344
7
xx
xx
x
This value is rejected. The solution set is
.
81. 2log log25x
2
2
log log 25
25
5
x
x
x
–5 is rejected. The solution set is {5}.
83. log( 4) log 2 log(5 1)xx
4
log log(5 1)
2
xx
84. log( 7) log 3 log(7 1)xx
7
20 4
1
5
xx
x
x
The solution set is
1
5
.
2
112
7
784
28
x
x
–28 is rejected. The solution set is {28}.
86. log( 2) log 5 log100x
87. log log( 3) log10xx
2
2
2
log( 3 ) log10
310
xx
xx
Section 12.4 Exponential and Logarithmic Equations
89. ln( 4) ln( 1) ln( 8)xxx
2
ln( 3 4) ln( 8)
xx x
90.
22 2
1
log ( 1) log ( 3) logxx x
22
11
log log
3
x
xx
91.
24
24 3
63
55 125
55
55
xx
xx
x
2
92.
2
24
3381
33
xx
xx
93.
2
2
345
ln 3
x
x
The solution set is ln 45 1.86
ln 3
.
2
x
The solution set is ln 50 1.56
ln 5
.
2
2
2
2
10 24
log 2
10 24 2
xx
x
xx
Chapter 12 Exponential and Logarithmic Functions
96.
222
log 3 log log 2 2xxx
22
log 3 log 2 2
xx x
97.
2
2
2
12 2
2
12 2
12 4
525
55
55
xx
x
x
xx
98.
2
2
2
12 2
2
12 2
12 4
39
33
33
xx
x
x
xx
99. a. 2010 is 0 years after 2010.
0.0095
37.3
t
Ae
The population of Texas was 25.1 million in
2010.
b.
0.0187
25.1
t
Ae
0.0187
28 25.1
0.0187
t
e
t
The population of Texas will reach 27 million
about 6 years after 2010, or 2016.
Section 12.4 Exponential and Logarithmic Equations
101. ( ) 20(0.975)
x
fx
1 20(0.975)
x
102. ( ) 20(0.975)
x
fx
3 20(0.975)
30.975
x
x
103.
4
0.0575
20000 12500 1 4
t
4
4
4
20000 12500 1 0.014375
20000 12500 1.014375
20000 1.014375
t
t
t
104. 1
nt
r
AP n
ln 12 ln 1.005417
29
60
ln 29 11.2
12ln 1.005417
t
t
ln1.4 720ln 1 360
ln1.4 ln 1
720 360
r
r
ln1.4 ln 1 360
720
r
ee
Chapter 12 Exponential and Logarithmic Functions
106.
360 4
1
9000 5000 1 360
nt
r
AP n
r
9
ln 1440ln 1 360
5
9
ln 5ln 1
1440 360
r
r
107.
0.08
0.08
0.08
0.08
16000 8000
16000
8000
2
ln 2 ln
t
t
t
t
e
e
e
e
108.
2
2
2
12000 8000
12000
8000
1.5
r
r
r
e
e
e
7
7
7
7050
2350
3
ln 3 ln
r
r
r
e
e
e
ln ln
697
e
1000
ln 0.0425
697
1000
ln 697 8.5
0.0425
t
t
Section 12.4 Exponential and Logarithmic Equations
b.
( ) 1.2ln 15.7fx x
18.5 1.2ln 15.7
x
112. a. 2008 is 2 years after 2006.
( ) 1.2ln 15.7
fx x
b.
( ) 1.2ln 15.7fx x
18.6 1.2ln 15.7
2.9 1.2ln
x
x
113.
2
50 95 30 log
x
114.
2
2
95 30log
09530log
Px x
x
115. a.
pH log
5.6 log
x
x
b.
pH log
2.4 log
2.4 log
x
x
x
116. a.
pH log
2.3 log
x
x
Chapter 12 Exponential and Logarithmic Functions
c.
1
1 ( 2.3) 1.3
2.3
10 10 10
117. –122. Answers will vary.
123.
1
28
x
124.
1
39
x
The solution set is
1.
125.
3
log 4 7 2x
The solution is 4, and the solution set is {4}.
126.
3
log 3 2 2x
Since
211
3,
33
the solution set is
11 .
3
Verify the solution algebraically:
127.
log 3 log 1
xx
The solution is 2, and the solution set is {2}.
log 5 2 1
log10 1
11
log 5 log 20 2
log 5 20 2
The solution set is
1.39,1.69
.
Verify the solutions algebraically:
x
6.40 6.38
The difference is due to rounding error.
130.
534
x
x
The solution set is
1.29,1.28 .
Section 12.4 Exponential and Logarithmic Equations
131.
0.48ln 1 27
fx x
132.
29 0.48ln 1 27
x
133.
0.092
145
t
Pt e
145 70
e
134.
3
0.075
2600 1 0.51
t
Wt e
135. does not make sense; Explanations will vary.
Sample explanation:
215
x
requires logarithms.
216
x
can be solved by rewriting 16 as
4
2.
215
ln 2 ln15
x
x
136. does not make sense; Explanations will vary.
Sample explanation: The first equation is solved by
137. makes sense
138. makes sense
139. false; Changes to make the statement true will vary.
142. false; Changes to make the statement true will vary.
A sample change is:
10
5.71
x
is not an
exponential equation, because there is not a variable
in an exponent.
Chapter 12 Exponential and Logarithmic Functions
143.
1
4000
0.03
4000 1 1
t
A
1
2000
0.05
2000 1 1
t
A
Set the right hand sides of the equations equal and
solve for t.
144.
22
2
ln ln
ln 2 ln
xx
xx
Let
ln
ux
.
2
2
uu
145.
log 2 log 1 6
xx
Let log .ux
2
2
216
26
260
23 20
uu
uu
uu
uu
xe
This value checks, so the solution set is
.
e
147.
21 11
xx
Section 12.5 Exponential Growth and Decay; Modeling Data
148.
2
35 19
1
35 19
11
xx
xx
xxxx
149.
44
328
4
32
2312
2
2216
xyy
xy yxx
150.
0.003
10
t
Ae
a. 2006:
0.003(0)
10 10 million
Ae
151. a.
ln 3
3
e
152. An exponential function is the best choice.
12.5 Check Points
1. a.
0
807
A
. Since 2011 is 11 years after 2000,
when
11
t
,
1052.
A
ln 807 0.024
11
k
Thus, the growth function is
0.024
807
t
Ae
.
b.
0.024
807
t
Ae
0.024
0.024
2000 807
2000
2000
ln 807 38
t
t
e
28
0
0
28
28
2
1
2
1
ln ln
k
k
k
AAe
e
e
Chapter 12 Exponential and Logarithmic Functions
b.
0.0248
0
t
AAe
0.0248
0.0248
10 60
10
t
t
e
e
3. A logarithmic function would be a good choice for
modeling the data.
4. An exponential function would be a good choice for
modeling the data although model choices may vary.
5. a. 1970 is 21 years after 1949.
( ) 0.074 2.294
(21) 0.074(21) 2.294
fx x
f
b. 2050 is 101 years after 1949.
( ) 0.074 2.294
(101) 0.074(101) 2.294
9.8
fx x
f
(ln 7.8)
2.054
4
4
x
x
ye
e
12.5 Concept and Vocabulary Check
3. logarithmic
4. exponential
5. linear
6.
ln 5
1. Since 2010 is 0 years after 2010, find A when
0t
:
0.006
0.006 0
127.3
127.3
t
Ae
Ae
Section 12.5 Exponential Growth and Decay; Modeling Data
2. Since 2010 is 0 years after 2010, find A when
0t
:
0.019
31.5
t
Ae
3. Since
0.019k
, Iraq has the greatest growth rate at
1.9% per year.
5. Substitute
1377A
into the model for India and
solve for t:
0.008
0.008
1377 1173.1
1377
1173.1
t
t
e
e
6. Substitute
1491A
into the model for India and
solve for t:
0.008
1491 1173.1
t
e
7. a.
0
6.04A
. Since 2050 is 50 years after 2000,
when
50t
,
10A
.
6.04
10
ln 50
k
b.
0.01
96.04
t
e
0.01
0.01
9
6.04
9
ln ln
6.04
t
t
e
e
0
when
50t
,
12A
.
0
(50)
12 3.2
kt
k
AAe
e
Chapter 12 Exponential and Logarithmic Functions
b.
0.026
3.2 9
t
e
0.026
9
3.2
t
e
9.
0.0095
0.0095(40)
0.0095(40)
() 99.9
(40) 99.9
(40) 99.9 146.1
t
Px e
Pe
Pe
11.
() 44.2
kt
Px e
40
40
62.9 44.2
62.9
44.2
k
k
e
e
12.
( ) 21.3
kt
Px e
40
42.7 21.3
0.0174
k
e
k
The growth rate is 0.0174.
13.
() 82.3
kt
Px e
40
70.5 82.3
ln 82.3
40
0.0039
k
e
k
k
The growth rate is –0.0039.
Section 12.5 Exponential Growth and Decay; Modeling Data
14.
() 7.1
kt
Px e
40
40
5.4 7.1
5.4
7.1
k
k
e
e
15.
0.000121
0.000121 5715
16
t
Ae
16.
0.000121
0.000121 11430
1.38303
16
16
16
4.01
t
Ae
Ae
Ae
A
Approximately 4 grams of carbon-14 will be present
in 11,430 years.
17. After 10 seconds, there will be
1
16 8
2
grams
present. After 20 seconds, there will be
1
84
18. After 25,000 years, there will be
1
16 8
2
grams
present. After 50,000 years, there will be
1
84
2
0.000121
0.000121
15
100
t
t
e
20.
0.000121
0
0.000121
0.000121
88 100
88
100
t
t
t
AAe
e
e
0.000121
ln 0.88 ln
ln 0.88 0.000121
ln 0.88 1056
0.000121
t
e
t
t
In 1989, the skeletons were approximately 1056
Chapter 12 Exponential and Logarithmic Functions
22.
0.063
0.5
0.5
kt
t
e
e
23.
1620
0.5
0.5
kt
k
e
e
24.
4560
0.5
0.5
kt
k
e
e
25.
17.5
0.5
0.5
kt
k
e
e
26.
113
0.5
0.5
kt
k
e
e
27. a.
1.31
11
2
k
e
The exponential model is given by
0.52912
0
t
AAe
.
b.
0.52912
0.52912
t
AAe
t
The age of the dinosaur bones is approximately
0.1069 billion or 106,900,000 years old.
ln 0.5
7340
0.000094
k
k
ln 0.2 0.000094
ln 0.2
0.000094
t
t