08-035_08_AFSM_C08_001-026.qxd
7/19/08
8:41 PM
Page 16
2x(0.699) 5 0.301 1 3x(0.602)
1.398x 5 0.301 1 1.806x
x 5 20.737
Substitute to determine y
y 5 3(5(23 (20.737)) )
y 5 3(521.475 )
y 5 0.279
log 63x 5 log 42x23
17. a)
3x log 6 5 (2x 2 3) log 4
3x(0.778) 5 (2x 2 3)(0.602)
2.33x 5 1.204x 2 1.806
1.13x 5 21.806
x 5 21.60
b) log (1.2)x 5 log (2.8)x14
x log 1.2 5 (x 1 4) log 2.8
0.079x 5 (x 1 4)(0.447)
0.079x 5 0.447x 1 1.789
20.368x 5 1.789
x 5 24.86
c)
log 3(2)x 5 log 4x11
log 3 1 log 2x 5 (x 1 1) log 4
log 3 1 x log 2 5 (x 1 1) log 4
0.477 1 0.301x 5 (x 1 1)(0.602)
0.477 1 0.301x 5 0.602x 1 0.602
20.125 5 0.301x
x 5 20.42
18. (2x )x 5 10
2
2x 5 10
2
log 2x 5 log 10
x2 log 2 5 1
1
x2 5
log 2
1
Å log 2
x 5 61.82
x56
8.6 Solving Logarithmic Equations,
pp. 491–492
1. a) log2 x 5 log2 52
x 5 25
b) log3 x 5 log3 34
x 5 81
c) log x 5 log 23
x58
d) log (x 2 5) 5 log 10
x 2 5 5 10
x 5 15
8-16
e) 2x 5 8
x53
f) log2 x 5 log2 !3
x 5 !3
4
2. a) x 5 625
x 4 5 54
x55
1
b) x2 2 5 6
1
56
!x
1 5 6 !x
1
5 !x
6
1
5x
36
c) 52 5 2x 2 1
25 5 2x 2 1
26 5 2x
x 5 13
d) 103 5 5x 2 2
1000 5 5x 2 2
1002 5 5x
x 5 200.4
e) x 22 5 0.04
1
5 0.04
x2
1 5 0.04x 2
25 5 x 2
x55
f) 2x 2 4 5 36
2x 5 40
x 5 20
a
1 4.2
3. 6.3 5 log
1.6
a
2.1 5 log
1.6
a
2.1
10 5
1.6
a
125.89 5
1.6
201.43 5 a
3
4. a) x2 5 33
x 3 5 (33 )2
x 3 5 (32 )3
x 3 5 93
x59
2
b) x 5 5
x 5 !5
Chapter 8: Exponential and Logarithmic Functions
08-035_08_AFSM_C08_001-026.qxd
7/23/08
10:48 AM
c) 33 5 3x 1 2
27 5 3x 1 2
25 5 3x
25
x5
3
d) 104 5 x
x 5 10 000
1 x
e) a b 5 27
3
32x 5 33
2x 5 3
x 5 23
1 22
f) a b 5 x
2
x54
5. a) log2 3x 5 3
23 5 3x
8 5 3x
8
x5
3
b) log 3x 5 1
10 5 3x
10
x5
3
c) log5 2x 1 log5 !9 5 2
log5 3 (3)(2x)4 5 2
log5 6x 5 2
52 5 6x
25 5 6x
25
x5
6
x
d) log4 5 2
2
x
42 5
2
x
16 5
2
x 5 32
e) log x 3 2 log 3 5 log 9
x3
5 log 9
log
3
x3
59
3
x 3 5 27
x 3 5 33
x53
Advanced Functions Solutions Manual
Page 17
f) log3 c
(4x)(5)
d 54
2
log3 10x 5 4
34 5 10x
81 5 10x
x 5 8.1
6. log6 3x(x 2 5)4 5 2
log6 (x 2 2 5x) 5 2
62 5 x 2 2 5x
36 5 x 2 2 5x
2
x 2 5x 2 36 5 0
(x 2 9)(x 1 4) 5 0
x 5 9 or x 5 24
Restrictions: x . 5 (x 2 5 must be positive) so x 5 9
7. a) log7 3 (x 1 1)(x 2 5)4 5 1
log7 (x 2 2 4x 2 5) 5 1
71 5 x 2 2 4x 2 5
x 2 2 4x 2 12 5 0
(x 2 6)(x 1 2) 5 0
x 5 6 or x 5 22
As x must be . 5, 22 is inadmissible. x 5 6
b) log3 3 (x 2 2)x4 5 1
log3 (x 2 2 2x) 5 1
31 5 x 2 2 2x
2
x 2 2x 2 3 5 0
(x 2 3)(x 1 1) 5 0
x 5 3 or x 5 21
As x must be . 2, 21 is inadmissible. x 5 3
x
c) log6
51
(x 2 1)
x
61 5
(x 2 1)
6x 2 6 5 x
5x 5 6
6
x5
5
d) log 3(2x 1 1)(x 2 1)4 5 log 9
2x 2 2 x 2 1 5 9
2x 2 2 x 2 10 5 0
(2x 2 5)(x 1 2) 5 0
x 5 2.5 or x 5 22
As x must be . 1, 22 is inadmissible. x 5 2.5
e) log 3(x 1 2)(x 2 1)4 5 1
101 5 x 2 1 x 2 2
x 2 1 x 2 12 5 0
(x 1 4)(x 2 3) 5 0
x 5 24 or x 5 3
As x must be . 1, 24 is inadmissible. x 5 3
8-17
08-035_08_AFSM_C08_001-026.qxd
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Page 18
f) log2 x 3 2 log2 x 5 8
x3
log2
58
x
log2 x 2 5 8
28 5 x 2
4 2
(2 ) 5 x 2
x 5 24 5 16
8. a) Use the rules of logarithms to obtain
log9 20 5 log9 x. Then, because both sides of the
equation have the same base, 20 5 x.
x
b) Use the rules of logarithms to obtain log 5 3.
2
Then use the definition of a logarithm
x
x
to obtain 10 3 5 ; 1000 5 ; 2000 5 x.
2
2
c) Use the rules of logarithms to obtain
log x 5 log 64. Then, because both sides of the
equation have the same base, x 5 64.
I
9. a) 50 5 10 log a 212 b
10
I
5 5 log a 212 b
10
5 5 log I 2 log 10212
5 5 log I 2 (212)
27 5 log I
1027 5 I
I
b) 84 5 10 log a 212 b
10
I
8.4 5 log a 212 b
10
8.4 5 log I 2 log 10212
8.4 5 log (I 1 12)
23.6 5 log I
1023.6 5 I
x12
d 5 loga (8 2 2x)
10. loga c
x21
x12
5 8 2 2x
x21
x 1 2 5 22x 2 1 10x 2 8
2
22x 1 9x 2 10 5 0
2x 2 2 9x 1 10 5 0
(2x 2 5)(x 2 2) 5 0
x 5 2.5 or x 5 2
11. a) x 5 0.80
b) x 5 26.91
c) x 5 3.16
d) x 5 0.34
8-18
12. log5 3 (x 2 1)(x 2 2)4 5 log5 (x 1 6)
(x 2 1)(x 2 2) 5 x 1 6
x 2 2 3x 1 2 5 x 1 6
x 2 2 4x 2 4 5 0
Using the quadratic formula
4 6 !16 2 (4)(24)
x5
2
4 6 !32
x5
2
x 5 4.83 or x 5 20.83
As x must be . 2, 20.83 is extraneous; x 5 4.83
13. log3 (28) 5 x; 3x 5 28; Raising positive 3 to
any power produces a positive value. If x $ 1, then
3x $ 3. If 0 # x , 1, then 1 # x , 3. If x , 0, then
0 , x , 1.
14. a) x . 3
b) If x is 3, we are trying to take the logarithm of 0.
If x is less than 3, we are trying to take the logarithm
of a negative number.
15. 12 (log x 1 log y) 5 12 log xy 5 log !xy so
x1y
5 !xy
5
and x 1 y 5 5!xy. Squaring both
sides gives (x 1 y)2 5 25xy. Expanding gives
x 2 1 2xy 1 y2 5 25xy; therefore, x2 1 y2 5 23xy.
16. log (35 2 x 3 ) 5 3 log (5 2 x)
log (35 2 x 3 ) 5 log (5 2 x)3
35 2 x 3 5 (5 2 x)3
35 2 x 3 5 2x 3 1 15x 2 2 75x 1 125
15x 2 2 75x 1 90 5 0
x 2 2 5x 1 6 5 0
(x 2 3)(x 2 2) 5 0
x 5 3 or x 5 2
17. log2 a 1 log2 b 5 4; log2 ab 5 4; 24 5 ab;
16 5 ab. The values of a and b that satisfy the
original equation are pairs that have a product of 16,
but a and b must also both be positive. The possible
pairs are: 1 and 16, 2 and 8, 4 and 4, 8 and 2, and
16 and 1.
18. log2 (5x 1 4) 5 3 1 log2 (x 2 1)
log2 (5x 1 4) 2 log2 (x 2 1) 5 3
5x 1 4
53
log2
x21
5x 1 4
23 5
x21
8(x 2 1) 5 5x 1 4
8x 2 8 5 5x 1 4
3x 5 12
x54
Chapter 8: Exponential and Logarithmic Functions
7/19/08
8:41 PM
Page 19
Substituting 4 in for x in the first equation
y 5 log2 ((5)(4) 1 4)
y 5 log2 24
2y 5 24
log 2y 5 log 24
y log 2 5 log 24
0.301y 5 1.38
y 5 4.58
19. a) 50 5 log3 x
log3 x 5 1
31 5 x
x53
b) 21 5 log4 x
log4 x 5 2
42 5 x
x 5 16
20. (221 )x1y 5 24
(x 2 y)23 5 8
1 23
22x2y 5 24
(x 2 y)23 5 a b
2
1
x2y5
2x 2 y 5 4
2
Adding the two equations gives
1
22y 5 4
2
y 5 22.25
Substituting into the first equation
2x 1 2.25 5 4
2x 5 1.75
x 5 21.75
8.7 Solving Problems with Exponential
and Logarithmic Functions, pp. 499–501
1. First earthquake: 5.2 5 log x; 105.2 5 158 489
Second earthquake; 6 5 log x; 106 5 1 000 000
Second earthquake is 6.3 times stronger than the
first.
2. pH 5 2log (H 1 )
pH 5 2log 6.21 3 1028
pH 5 2 (27.2)
pH 5 7.2
3. 1 000 000 3 10212 W>m2 5 1026 W>m2; the
intensity of the sound
1026
L 5 10 log 212
10
L 5 10 log 106
L 5 (10)(6) 5 60 dB
Advanced Functions Solutions Manual
I
I
60 5 10 log 212
212
10
10
6 5 log I 2 log 10212
6.9 5 log I 2 log 10212
6.9 5 log I 1 12
6 5 log I 1 12
25.1 5 log I
26 5 log I
1025.1 5 I
1026 5 I
I 5 1 3 1026
I 5 7.9 3 1026
A heavy snore is 7.9 times louder than a normal
conversation.
5. a) 9 5 2log H
29 5 log H
1029 5 0.000 000 001 5 H
b) 6.6 5 2log H
26.6 5 log H
1026.6 5 0.000 000 251 5 H
c) 7.8 5 2log H
27.8 5 log H
10 27.8 5 0.000 000 016 5 H
d) 13 5 2log H
213 5 log H
10 213 5 0.000 000 000 000 1 5 H
6. a) pH 5 2log 0.000 32 5 3.49
b) pH 5 2log 0.000 3 5 3.52
c) pH 5 2log 0.000 045 5 4.35
d) pH 5 2log 0.005 5 2.30
7. a) pH 5 2log 1027 5 7
b) Tap water is more acidic than distilled water as
it has a lower pH than distilled water (pH 7).
I
I
118 5 10 log 212
8. 109 5 10 log 212
10
10
10.9 5 log I 2 log 10212 11.8 5 log I 2 log 10212
10.9 5 log I 1 12
11.8 5 log I 1 12
21.1 5 log I
20.2 5 log I
1021.1 5 I
1020.2 5 I
I 5 0.079
I 5 0.63
An amplifier is 7.98 times louder than a lawn mower.
9. a) y 5 5000(1.0642)t
Investment Growth
12 000
4. 69 5 10 log
10 000
Amount ($)
08-035_08_AFSM_C08_001-026.qxd
8000
6000
4000
2000
0
2
4
6 8 10
Year
8-19