08-035_08_AFSM_C08_001-026.qxd
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8:41 PM
function: y 5 20(8)x
function: y 5 25x 2 3
D 5 5xPR6
R 5 5yPR0 y , 236
asymptote: y 5 23
inverse: y 5 log 8 Q R
20
inverse:
y 5 log5 (2x 2 3)
D 5 5xPR0 x , 236
R 5 5yPR6
asymptote: x 5 23
D 5 5xPR6
R 5 5yPR0 y . 06
asymptote: y 5 0
x
D 5 5xPR0 x . 06
R 5 5yPR6
asymptote: x 5 0
e)
Page 9
23. Given the constraints, two integer values are
possible for y, either 1 or 2. If y 5 3, then x must
be 1000, which is not permitted.
y
16
12
8
y 52(3)x+2
4
x
–8 –4 0 4 8 12 16
–4
x
–8 y 5log3 (—
2 ( –2
function: y 5 2(3)x12
D 5 5xPR6
R 5 5yPR0 y . 06
asymptote: y 5 0
inverse: y 5 log 3 Q R 22
2
x
D 5 5xPR0 x . 06
R 5 5yPR6
asymptote: x 5 0
y
f)
2
–10 –8 –6 –4 –2 0
–2
y 525x23 –4
–6
–8
–10
y5log5(2x23) –12
–14
x
2
Advanced Functions Solutions Manual
8.4 Laws of Logarithms, pp. 475–476
1. a) log 45 1 log 68
b) logm p 1 logm q
c) log 123 2 log 31
d) logm p 2 logm q
e) log2 14 1 log2 9
f) log4 81 2 log4 30
2. a) log (7 3 5) 5 log 35
4
b) log3 5 log3 2
2
c) logm ab
x
d) log
y
e) log6 (7 3 8 3 9) 5 log6 504
(10 3 12)
f) log4 a
b 5 log4 6
20
3. a) 2 log 5
b) 21 log 7
c) q logm p
1
d) log 45
3
1
e) log7 36
2
1
f) log5 125
5
135
4. a) log3 135 2 log3 5 5 log3
5
5 log3 27
53
b) log5 10 1 log5 2.5 5 log5 (10 3 2.5)
5 log5 25
52
8-9
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c) log 50 1 log 2 5 log (50 3 2)
5 log 100
52
d) log4 47 5 7 log4 4
5731
57
224
e) log2 224 2 log2 7 5 log2
7
5 log2 32
55
1
f) log !10 5 log 10
2
1
5 a b (1)
2
1
5
2
5. y 5 log2 (4x) 5 log2 x 1 log2 4 5 log2 x 1 2, so
y 5 log2 (4x) vertically shifts y 5 log2 x up 2 units;
y 5 log2 (8x) 5 log2 x 1 log2 8 5 log2 x 1 3, so
y 5 log2 (8x) vertically shifts y 5 log2 x up 3 units;
y 5 log2 Q R 5 log2 x 2 log2 2 5 log2 x 2 1, so
2
x
y 5 log2 Q R vertically shifts y 5 log2 x down 1 unit
2
x
6. a) log25 53 5 3 log25 5
log25 5; 25x 5 5; x 5 0.5
Therefore log25 53 5 3 log25 5 5 (3)(0.5) 5 1.5
(54 3 2)
b) log6 54 1 log6 2 2 log6 3 5 log6
3
5 log6 36
52
c) log6 6 !6 5 log6 6 1 log6 !6
5 1 1 0.5
5 1.5
!36
d) log2 !36 2 log2 !72 5 log2
!72
1
5 log2
Å2
5 log2 220.5
5 20.5
3
e) log3 54 1 log3 a b 5 log3 54 1 log3 3 2 log3 2
2
5 log3 54 2 log3 2 1 1
54
11
5 log3
2
5 log3 27 1 1
5311
54
8-10
f) log8 2 1 3 log8 2 1
1
log8 16
2
5 log8 2 1 log8 23 1 log8 !16
5 log8 2 1 log8 8 1 log8 4
5 log8 2 1 log8 4 1 1
5 log8 (2 3 4) 1 1
5 log8 8 1 1
5111
52
7. a) logb x 1 logb y 1 logb z
b) logb z 2 (logb x 1 logb y)
c) logb x 2 1 logb y 3 5 2 logb x 1 3 logb y
1
d) logb "x 5yz 3 5 logb x 5yz 3
2
1
5 (logb x 5 1 logb y 1 logb z 3 )
2
1
5 (5 logb x 1 logb y 1 3 logb z)
2
8. log5 3 means 5x 5 3 and log5 13 means 5y 5 13;
since 13 5 321, 5y 5 5x(21); therefore,
log5 3 1 log5 13 5 x 1 x(21) 5 0
9. a) 3 log5 2 1 log5 7 5 log5 23 1 log5 7
5 log5 8 1 log5 7
5 log5 (8 3 7)
5 log5 56
b) 2 log3 8 2 5 log3 2 5 log3 82 2 log3 25
5 log3 64 2 log3 32
64
5 log3
32
5 log3 2
c) 2 log2 3 1 log2 5 5 log2 32 1 log2 5
5 log2 9 1 log2 5
5 log2 (9 3 5)
5 log2 45
(12 3 2)
d) log3 12 1 log3 2 2 log3 6 5 log3
6
5 log3 4
1
e) log4 3 1 log4 8 2 log4 2
2
5 log4 3 1 log4 !8 2 log4 2
(3 !8)
2
5 log4 (3 !2)
f ) 2 log 8 1 log 9 2 log 36
5 log 82 1 log 9 2 log 36
5 log 64 1 log 9 2 log 36
(64 3 9)
5 log
36
5 log 16
5 log4
Chapter 8: Exponential and Logarithmic Functions
08-035_08_AFSM_C08_001-026.qxd
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10. a) log2 x 5 log2 72 1 log2 5
log2 x 5 log2 49 1 log2 5
log2 x 5 log2 (49 3 5)
log2 x 5 log2 245
x 5 245
b) log x 5 log 42 1 log 33
log x 5 log 16 1 log 27
log x 5 log (16 3 27)
log x 5 log 432
x 5 432
c) log4 x 5 log4 48 2 log4 12
48
log4 x 5 log4
12
log4 x 5 log4 4
x54
d) log7 x 5 log7 252 2 log7 53
log7 x 5 log7 625 2 log7 125
625
log7 x 5 log7
125
log7 x 5 log7 5
x55
e) log3 x 5 log3 102 2 log3 25
log3 x 5 log3 100 2 log3 25
100
log3 x 5 log3
25
log3 x 5 log3 4
x54
f) log5 x 5 log5 6 1 3 log5 2 1 log5 8
log5 x 5 log5 6 1 log5 23 1 log5 8
log5 x 5 log5 6 1 log5 8 1 log5 8
log5 x 5 log5 (6 3 8 3 8)
log5 x 5 log5 384
x 5 384
11. a) log2 xyz
uw
b) log5 v
a
c) log6 a 2 log6 bc 5 log6
bc
x 2y 2
5 log2 xy
d) log2
xy
e) log3 3 1 log3 x 2 5 log3 3x 2
x 3x 2
f) log4 x 3 1 log4 x 2 2 log4 y 5 log4
y
x5
5 log4
y
1
1
12. loga x 5 loga !x; loga y 5 loga !y;
2
2
3
4 3
loga z 5 loga "
z;
4
Advanced Functions Solutions Manual
1
1
3
loga x 1 loga y 2 loga x
2
2
4
4 3
5 loga !x 1 loga !y 2 loga "
z
So,
5 loga
!x !y
4 3
"
z
13. vertical stretch by a factor of 3 (the exponent
that x is raised to); vertical shift 3 units up (the
coefficient 8 divided by the base of 2)
14. Answers may vary. For example,
f(x) 5 2 log x 2 log 12
x2
g(x) 5 log
12
2 log x 2 log 12 5 log x 2 2 log 12
x2
5 log
12
So, f(x) and g(x) have the same graph.
15. Answers may vary. For example, any number
can be written as a power with a given base. The
base of the logarithm is 3. Write each term in the
quotient as a power of 3. The laws of logarithms
make it possible to evaluate the expression by
simplifying the quotient and noting the exponent.
16. logx x m21 1 1 5 m 2 1 1 1 5 m
17. logb x !x 5 logb x 1 logb !x
1
5 logb x 1 logb x
2
1
5 0.3 1 0.3a b
2
5 0.45
18. The two functions have different domains. The
first function has a domain of x . 0. The second
function has a domain of all real numbers except 0,
since x is squared.
19. Answers may vary; for example,
Product law
log10 10 1 log10 10 5 1 1 1
52
5 log10 100
5 log10 (10 3 10)
Quotient law
log10 10 2 log10 10 5 1 2 1
50
5 log10 1
10
5 log10 a b
10
Power law
log10 102 5 log10 100
52
5 2 log10 10
8-11