Section 12.2 Logarithmic Functions
102.
ln ln 3fx x gx x
103.
ln ln 3fx x gx x
104.
log logfx x gx x
105.
log log 2 1fx x gx x
106.
75 10log( 1)ft t
falls below 65.
107. a.
ln 3
ln 3 ln
fx x
gx x
The graphs coincide.
b.
2
2
log 5
log 5 log
fx x
gx x
The graphs coincide.
Chapter 12 Exponential and Logarithmic Functions
c.
3
3
ln 2
ln 2 ln
fx x
gx x
d. In each case, the function, f, is equivalent to g.
This means that
log log log .
bbb
MN M N
108.
109. makes sense
110. does not make sense; Explanations will vary.
Sample explanation: Logarithmic functions do not
113. false; Changes to make the statement true will vary.
A sample change is:
2
2
log 8 3
log 4 2
114. false; Changes to make the statement true will vary.
A sample change is: We cannot take the log of a
negative number.
117. To evaluate
3
22
log 81 log 1
log 8 log 0.001
π
, consider each of
the terms independently.
4
33
4
y
y
0
0
y
y
ππ
2
32
3
y
3
22
log 81 log 1 40 4 4
log 8 log0.001 2 3 2 3 5
π
Section 12.3 Properties of Logarithms
119. To determine which expression represents a greater
number, rewrite the expressions in exponential
notation.
120. Rewrite the equations in
A
xByC
form.
25 11
32 12
xy
xy
Multiply the first equation by 2 and the second
equation by 5 and solve by addition.
121.
2222
682 234
23
xxyy xxyy
xyxy
123. a.
5
22
log 32 log 2 5
2
125. a.
4
33
log 81 log 3 4
b.
2
33
2log 9 2log 3 2 2 4
2
b.
ln ln ln11
11
5ln11
Chapter 12 Exponential and Logarithmic Functions
b.
1
2
55
33
log log
25 25
xx
yy
5. a.
log 25 log 4 log(25 4)
log100
2
b.
76
log(7 6) log log
x
xxx
x
c.
1
4
1
4
210
1
4
210
10
4
10
4
log 2log 5 10log log log 5 log
log log 5 log
log log 25 log
log log 25
bb bbbb
bbb
bbb
bb
xyxy
xy
xy
xy
Section 12.3 Properties of Logarithms
12.3 Concept and Vocabulary Check
1.
log log
bb
MN
; sum
12.3 Exercise Set
1.
555
log 7 3 log 7 log 3
2.
888
log 13 7 log 13 log 7
3.
7777
log 7 log 7 log 1 log
xxx
9.
log log log100 log 2
100
xxx
10.
log log log1000 log 3
1000
xxx
13.
22
ln ln ln 5 2 ln 5
5
ee
17.
6
log 6log
NN
18.
8
log 8log
MM
19.
1
55
1
ln ln ln
5
xx x
1
1
4
1log 3
2
x
24.
555
1
2
log log log 25
25
xx
Chapter 12 Exponential and Logarithmic Functions
28.
332
2
3
log log log
log log 2log
3log log 2 log
bbb
bb b
bb b
xy xy z
z
xyz
xy z
30.
1
2
1
ln ln ln
2
11
ln ln 1 ln
22
11
ln
22
ex ex ex
ex x
x
1
2
33
log log log
1log 3log 3log
2
bb
bbb
bbb
xyz
xyz
1
3
44
3
xy x y
1
23
5
2
5
2
log 25
1log
325
1log log 25
xy
xy
Section 12.3 Properties of Logarithms
37.
log5 log 2 log 5 2 log10 1
38.
log 250 log 4 log 250 4 log1000 3
39.
ln ln 7 ln 7 ln 7
xxx
43.
25
log 2 5 log log
x
xxx
44.
37
log 3 7 log log
x
xx
x
2
y
52.
7
73
3
7ln 3ln ln ln ln
x
xyxy y
53.
1
33
1
3ln ln ln ln
xyxy
22
1
2
ln ln
xx
y
y
55.
43
4ln63lnln6ln
xxxx
Chapter 12 Exponential and Logarithmic Functions
59.
1
2
55 5
2
55
2
55
2
55
52
1log log 2log 1
2
1log log 1
2
log log 1
log log 1
log
1
xy x
xy x
xy x
xy x
xy
x
61.
5
log13
log 13 1.5937
log5
62.
6
log17
log 17 1.5812
log 6
63.
14
log87.5
log 87.5 1.6944
log14
69.
3
log log 3 log 2
2
bbb
CA
70.
log 6 log 2 3 log 2 log 3
bb bb
AC
71.
3
log 8 log 2 3log 2 3
bb b
A
72.
4
log 81 log 3 4 log 3 4
bb b
C
73.
1
2
22
log log
bb
74.
1
2
2
33
log log
16 4
log 3 log 4
log 3 log 2
1log 3 2 log 2
2
12
2
bb
bb
bb
bb
CA
Section 12.3 Properties of Logarithms
79. true
80. false; Changes to make the statement true will vary. A sample change is: ln( 1) ln ln1xx .
1
x
85. true
86. true
90. a.
2
log 16 4
b.
22 222
5
22 2
5
2
log 5log 4 log 5log log 16
log log log 16
log 16
xy xy
xy
xy
Chapter 12 Exponential and Logarithmic Functions
93. a.
0
0
10(log log )
10log
DII
I
I
94. a.
1ln ln
11
ln ln
tAAN
c
AA
cANcAN
95. – 102. Answers will vary.
Section 12.3 Properties of Logarithms
103. a.
3
log
log log 3
x
yx
b.
3
3
3
2log
log 2
log
yx
yx
yx
3
log
yx
The graph of
3
2log
yx
is the graph of
3
log
yx
shifted up two units.
104.
log
log(10 )
log(0.1 )
yx
yx
yx
The graph of
log 0.1
yx
is the graph of
logyx shifted down 1 unit.
The product rule accounts for this relationship.
Consider
log 10 .
yx
105.
3
25
100
log
log
log
yx
yx
yx
b.
3
log
yx
is on top.
100
log
yx
is on the
bottom.
graphed for two different values of b, the graph
of the one with the smaller base will be on the
bottom in the interval (0, 1) and the one with the
larger base will be on the bottom in the interval
1, .
106. – 110. Answers will vary
111. makes sense
Chapter 12 Exponential and Logarithmic Functions
114. does not make sense; Explanations will vary.
115. true
116. false; Changes to make the statement true will vary.
A sample change is:
77
7
7
log 49 log 49 log 49 2
log 7 1
, but
77
log 49 log 7 2 1 1
.
118. false; Changes to make the statement true will vary.
A sample change is:
5
log 5log
5log log
5log 5log
bb
bb
bb
xy xy
xy
xy
119. Recall that when a logarithm is written without a
base, the base is 10.
Find the x–intercept by setting y = 0.
Next, use the origin as a test point.
50 20 10
0010
010
This is a false statement. This means that the origin
will not fall in the shaded half-plane.
6423
5423
xx x
xx
74 3
77
1
x
x
x
The solution set is
,1 .
12
Mid-Chapter Check Point
127.
21
43
21
(4 3) (4 3)
43
x
xx
x
xx xx
xx
Mid-Chapter Check Point – Chapter 12
1.
11
4
5
2
23
22.75
12.5
x
xfx
2.
1
3
29
x
xfx
Domain:
,
10
21
42
Chapter 12 Exponential and Logarithmic Functions
4.
2
1
4
1
log 1
1
0
xfx x
5.
3
log 6fx x
The argument of the logarithm must be positive:
60
6
x
x
Domain:
6,.
7.
2
3
log 6x
The argument of the logarithm must be positive.
Now
2
6x is always positive, except when
6x .
Domain:
66, .
11. Let
100
21
log 10
100 10
2
y
y
y
y
.
12.
1
3
3
1
log 10 log10 3
3
32
3
82
log log 2
log 3
not possible
This expression is impossible to evaluate.
15. 6
log 5
65
18.
100
2
21
log 0.1
100 0.1
1
10 10
10 10
21
y
y
y
y
y
Section 12.4 Exponential and Logarithmic Equations
21.
19 20 19 20
ln ln ln
19 20ln
ex e x
x
23.
72
5555
72
5
9
5
7log 2log log log
log
log
xxxx
xx
x
25. Continuously:
0.08(3)
8000
10,170
Ae
Monthly:
12 3
0.08
8000 1 12
10,162
A
10,170 10,162 8
Interest returned will be $8 more if compounded
continuously.
25
25
22
22
x
x
5134
ln 5 ln134
ln 5 ln134
ln134
ln 5
x
x
x
3.90
x
The solution set is
log8000 3.90 .
Chapter 12 Exponential and Logarithmic Functions
3. Isolate the exponential expression then take the
natural log of both sides of the equation.
2
7558
x
e
4. a.
2
2
3
log ( 4) 3
log ( 4) 3
24
84
12
x
x
x
x
x
12 checks. The solution set is
12 .
5.
2
12
log log( 3) 1
log( 3 ) 1
10 3
xx
xx
xx
6. ln( 3) ln(7 23) ln( 1)xxx
723
ln( 3) ln 1
x
xx
12.77
12.77
7
6
7
ln ln 6
7
12.77 ln 6
7
ln 6
x
x
e
e
x
4
4
4
3.6 1.02
1.02 3.6
ln1.02 ln 3.6
t
t
t
Section 12.4 Exponential and Logarithmic Equations
9. ( ) 62 35log( 4)fx x
Solve the equation when () 97fx.
62 35log( 4) 97
x
12.4 Concept and Vocabulary Check
1. M = N
2.
41x
3.
ln 20
ln 9
12.4 Exercise Set
1.
6
264
22
6
x
x
x
3.
3
5125
55
x
x
5.
21
21 5
232
22
215
26
3
x
x
x
x
x
The solution set is
3.
2
x
The solution set is
2.
8.
31
31 3
5125
55
313
x
x
x
Chapter 12 Exponential and Logarithmic Functions
9.
32 8
x
10.
25
432
22
x
x
11.
23
927
33
x
x
12.
34
125 625
55
x
x
13.
1
1
327
x
The solution set is {4}.
14.
2
1
5125
x
15.
3
4
66
5
x
x
x
The solution set is {5}.
Section 12.4 Exponential and Logarithmic Equations
17.
1
2
2
1
42
1
2
2
x
x
4
18.
1
3
3
2
1
93
1
3
3
x
x
19.
5.7
ln ln 5.7
ln 5.7 1.74
x
x
e
e
x
The solution set is
ln 5.7 1.74 .
21.
10 3.91
log10 log 3.91
x
x
22.
10 8.07
log10 log8.07
log8.07
x
x
x
The solution set is ln17 1.76 .
ln 5
24.
19 143
ln19 ln143
x
x
ln ln 5
ln 5 1.61
x
e
x
The solution set is
ln 5 1.61 .
26.
999
11
x
x
e
e
27.
5
5
3 1977
659
ln ln 659
x
x
e
e
e
Chapter 12 Exponential and Logarithmic Functions
28.
7
7
410,273
2568.25
x
x
e
e
29.
0.7
0.7
13
ln ln13
0.7 ln13
ln13 3.66
0.7
x
x
e
e
x
x
The solution set is
ln13 3.66 .
0.7
31.
0.055
0.055
0.055
1250 3750
3
ln ln 3
0.055 ln 3
ln 3 19.97
0.055
x
x
x
e
e
e
x
x
The solution set is
ln 3 19.97 .
33.
30 1.4 0
1.4 30
x
x
34.
135 4.7 0
4.7 135
4.7 135
ln 4.7 ln135
ln 4.7 ln135
ln135 3.17
ln 4.7
x
x
x
x
x
x
5
1ln793 1.14
5
x
The solution set is 1ln793 1.14
5
.
36.
18
18
7957
ln ln 7957
x
x
e
e