Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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Special Topics 4.5.A, page 307
1. y x; 14 x 14 and 14 y 14
3. y 2x 3; 15 x 17 and 30 y 30
5. Vertical asymptote: x 2
Nonvertical asymptote: y x 1,
7. Vertical asymptote: x
5
2
Nonvertical asymptote: y 2x 1
9.
Vertical asymptote: x 1
Parabolic asymptote: y x
2
x 1
y
x
132
y
= x
2
+ x + 1
−2
1
4
6
8
10
12
−2
−4
−6
4 2264108
10
20
20
10
x
108 6
y
x
y
−6
−2
8
6
4
−4
−8
2
4
6
8
2
0
−2
−4
−6
−8
996 ANSWERS
11. Vertical asymptotes: 1, 2, 1
Nonvertical asymptote: y x
2
5x
13. 15.5 x 8.5 and 16 y 8
15. 7 x 14 and 30 y 30
17. Overall: 13 x 7 and 20 y 20; hidden area near
the origin: 2.5 x 1 and .02 y .02
19. (a)
(b) Approx. 0.06 x 2.78
21. y rx b
Section 4.6, page 315
1. (∞, 3/2) 3. (2, )
5. (, 8/5] 7. (1, )
9. (2, 4) 11. [3, 5/2)
13. (4/45, ) 15. [7/17, )
17. [1, 1/8) 19. [5, )
21. x
b
a
c
23. c x a c
25. 1 x 3
27. x
1
2
33
or x
1
2
33
29. 1 x 0 or x 1
31. 1 x 1 and x 3
33. 2 x 1 or 1 x 2
35. .5 x .84 37. x 1/3 or x 2
39. x 1 41. 3 x 1 or x 5
43. 7
x 7
or x 5.34
45. x 3 or 1/2 x 5 47. x 0 or x 1
49. x 1.43 51. x 3.79 or x .79
53. Approximately 8.608 cents per kwh
55. More than $37,500
112 2345
20
40
20
x
345
40
y
57. (a) They would have to talk more than 6 months.
(b) They are saving between 42 and 60 dollars per month,
not counting the up-front fees to switch.
59. 1 x 19 and y 20 x
61. 10 x 35 63. 1 t 4 65. 2 t 2.25
67. (a) x
2
x when 0 x 1 and x
2
x when x 0 or x 1.
(b) If c is nonzero and c 1, then either 0 c 1 or 1
c 0 (which is equivalent to 1 c 0). If 0 c
1, then c c and c is a solution of x
2
x by part (a),
so that c
2
c c. If 1 c 0, then c c,
which is a solution of x
2
x by part (a), so that c
2
(c)
2
(c) c.
(c) If c 1, then either c 1 or c 1. In either case,
c is a solution of x
2
x by part (a).
69. No solutions
71. (∞, ∞), all real numbers are solutions.
73. x 2, x 3 75. No solutions
77. x
5
2
Special Topics 4.6.A, page 320
1. 4/3 x 0 3. 7/6 x 11/6
5. x 11/20 or x 1/4
7. x 53/40 or x 43/40
9. x 7/2 or x 5/4 11. 1/7 x 3
13. 3
x 1 or 1 x 3
15. x 6
or x 6
17. x 2 or 1 x 0 or x 1
19. 1.43 x 1.24 21. x .89 or x 1.56
23. x 2 or x 14/3
25. 1.13 x 1.35 or 1.35 x 1.67
27. (∞, ∞), all real numbers are solutions.
29. x 1, x 2
31. If x 3 E/5, then multiplying both sides by 5 shows
that 5x 3 E. But
5x 3 5
x 3 5(x 3)
5x 15 (5x 4) 11. Thus,
(5x 4) 11 E.
33. (a) 2.999 x 3.001
(b) d l .001 where d is the desired length, and l is the
length of the manufactured rod.
Section 4.7, page 326
1. 8 2i 3. 2 10i
5. 1/2 2i 7.
2
2
3
2i
9. 1 13i 11. 30i
ANSWERS 997
13. 21 20i 15. 4
17. i 19. i
21. i 23.
3
13
2i
25.
4
3
i 27.
1
4
2
1
1
4
5
1
i
29.
4
5
1
4
4
1
i 31.
1
1
0
7
1
1
1
7
i
33.
1
7
0
1
1
1
0
i 35.
1
1
1
7
3
0
1
4
7
1
0
i
37. 6i 39. 14i
41. 4i 43. 11i
45. (15 32
)i 47. 2/3
49. 41 i
51. (2 52
) (5
210)i
53.
1
6
1
6
i 5
55. x 2, y 2
57. x 3/4, y 3/2 59.
1
3
3
14
i,
1
3
3
14
i
61. 2 0i, 3 0i 63.
1
4
4
31
i,
1
4
4
31
i
65. 42 2.5i, 42 2.5i
67. 2 0i, 1 3
i, 1 3
i
69. 1 0i, 1 0i, 0 i, 0 i
71. 1
73. (a) (i) 5 (ii) 25 (iii) 8
(iv) 8 (v) 8
(b) mod(5 12i)
75. z
w
(
a
b
i
)
(
c
d
i
)
(ac bd) (bc ad)
i
(ac bd) (bc ad)i
z
w
(a bi)
(c di)
(a bi)(c di)
(ac bd) (bc ad)i
Thus z
w
z
w
77. z
a b
i a b
i a bi z
79. (a)
z
2
z
2
a
a
a
(b)
z
2i
z
2
2
b
i
i
b
81. (a) (i) (a, b) (c, d) (a c, b d) (c a, d b)
(c, d) (a, b)
(ii) [(a, b) (c, d)] (e, f ) (a c, d d) (e, f )
((a c)) e, (b d) f )
(a (c e), b (d f ))
(a, b) [(c, d) (e, f )]
(iii) (a, b) (0, 0) (a 0, b 0) (a, b)
(iv) (a, b) (a, b) (a (a), b (b)) (0, 0)
(a bi) (a bi)
2i
(a bi) (a bi)
2
(b) (i) (a, b)(c, d) (ac bd, bc, ad )
(ca db, cb da) (c, d)(a, b)
(ii) [(a, b)(c, d)](e, f ) (ac bd, bc ad)(e, f )
(ace adf bcf bde,
acf ade bce bdf )
(a, b)(ce df, cf de)
(
a, b)[(c, d)(e, f )]
(iii) (a, b)(1, 0) (a
1 b
0, a
0 b
1) (a, b)
(iv) (a, b)(0, 0) (a
0 b
0, a
0 b
0) (0, 0)
(c) (i) (a, 0) (c, 0) (a c, 0 0) (a c, 0)
(ii) (a, 0)(c,0) (a
c 0
0, a
0 c
0) (ac,0)
(d) (i) (0, 1)(0, 1) (0
0 1
1, 0
1 0
1) (1, 0)
(ii) (b, 0)(0, 1) (b
0 0
1, b
1 0 0) (0, b)
(iii) (a,0) (b, 0)(0, 1) (a,0) (0, b) (a 0, 0 b)
(a, b)
Section 4.8, page 332
1. 3 3. 0 5. Yes 7. Yes
9. 0: multiplicity 54, 4/5: multiplicity 1
11. 0: multiplicity 15; p: multiplicity 14; p 1: multiplicity 13
13. 1 2i, [x (1 2i)][x (1 2i)]
15. 3 with multiplicity 2; 3(x 3)(x 3)
17. 3,
2
3
3
2
3
i,
2
3
3
2
3
i,
(x 3)
x
3
2
33
i
x
3
2
33
i
19. 2, 1 3
i,1 3
i,
(x 2)[x (1 3
i)][x (1 3
i)]
21. 1, i,(x 1)(x 1)(x i)(x i)
23. 1, 1 i,(x 1)(x 1)(x 1 i)(x 1 i)
In Exercises 25–42, there may be correct answers other than
the ones shown here.
25. (x 1)(x 7)(x 4) 27. (x 1)
2
(x 2)
2
(x p)
2
29. f (x) 2x(x 3)(x 4) 31. f (x) x
2
4x 5
33. f(x) x
5
x
4
x
3
19x
2
32x 30
35. f (x) x
2
2x 5
37. f (x) x
4
14x
3
74x
2
176x 160
39. f(x) x
6
5x
5
8x
4
6x
3
41. f (x) 3x
2
6x 6 43. x
2
(1 i) x (2 i)
45. x
3
5x
2
(7 2i) x (3 6i)
47. 3,
1
2
2
3
i,
1
2
2
3
i
49. i, i, 1, 2 51. 2 i, 2 i, i, i
53. (a) z
w
(
a
b
i
)
(
c
d
i
)
(
a
c
)
(
b
d
)
i
(a c) (b d)i
(a bi) (c di)
z
w
998 ANSWERS
(b) z
w (a bi)
(c di)
(ac bd) (bc ad)
i
(ac bd) (bc ad )i
(a bi)
(c di)
z
w
55. (a) If z is a root of f (x), f (z) 0. Then f
(
z
)
0
0. Also
f
(
z
)
a
z
3
b
z
2
c
z
d
az
3
bz
2
c
z
d
a
z
3
bz
2
c
z
d
az
3
bz
2
cz
d f (z
)
(b) Since f
(
z
)
0, az
3
bz
2
cz
d 0; therefore, z
is
a root of f (x).
57. For each nonreal complex root z, there must be two factors
of the polynomial: (x z) and (x z
). This yields an even
number of factors. There will remain at least one factor,
hence at least one root, which must be real.
Chapter 4 Review, page 335
1. (2, 3) 3. (4, 4)
5. (1.5, 5.75)
7. (a) y 260 x
(b) A(x) x
2
260x 3500
(c) 130 ft by 130 ft
9. 30 ft perpendicular to the building by 60 ft parallel to the
building
11. (a), (c), (d), and (f) 13. 0
15. 2158117 16 4
2 64 1014 4
1 325720
the other factor is x
5
3x
4
2x
3
5x
2
7x 2
17. f (x) x
3
5x
2
x 5 is one of many correct answers
19. 5/3 and 1 21.
3
2
23. 0,
1
2
21
25. (a) 1, 1, 3, 3,
1
2
,
1
2
,
3
2
,
3
2
(b) 3 (c) x
1
2
3
and x 3
27. 1
29. 3, 3, 2
, 2
31. (x
4
4x
3
16x 16) (x 5) is x
3
x
2
5x 41 with
remainder 189. Since all coefficients and the remainder are
positive, 5 is an upper bound for the roots.
33. 1, 1, 1.867, .867
35. 2x h 1
37.
This graph could not possibly be the graph of a polynomial
function because of the sharp corner. Many other graphs are
possible.
39. (c)
41. 1 x 2 and 10 y 60
43. 2 x 18 and 500 y 1200
45. (a)
(b) y .00852x
3
1.10945x
2
56.2583x 2017.2576
(c) C(71) C(70) $26.79
(d) About $85.49, about $47.84
47.
x-intercepts: 3, 0, 3; local maximum: (1.732, 10.392);
local minimum: (1.732, 10.392)
49.
x-intercepts: (4, 0), (3, 0)(2, 0); y-intercept: (0, 48);
local minima when x 3.54 and x 2; local maximum
when x .71
21
x intercept
local min
local max
x intercept
y intercept
x intercept
and local min
24
10
10
x
13453
20
30
40
50
60
70
80
90
100
y
x
y
2
4
15
10
5
−4
−2
−5
−10
−15
1757.75
3667.25
−777
x
1
2
3
5
4
3
2
1
−2
−1
f(x)
ANSWERS 999
51.
x-intercepts: none; vertical asymptote: x 4; horizontal
asymptote: y 0
53.
x-intercept: 2.5; vertical asymptote: x 3; horizontal
asymptote: y 4/3
55. vertical asymptotes: x 2, x 3; horizontal asymptote:
y 0
57. 4.7 x 4.7 and 5 y 5; then 2 x 20 and
.2 y .1
59. 18.8 x 18.8 and 8 y 8
61. between 400 and 2500 bags; between $1.75 (at 400 bags)
and $0.70 (at 2500 bags)
63. (a) 0.455 s 5.498
(b) 1.581
(c) 10 miles/hour. This is a very low speed limit, and the
citizens would probably object.
65.
(x 1)(x
1
h 1)
67. (d)
69. y 17 or y 13
71. (∞, 2) and (1/3, ∞)
73. 3
x 1 or 1 x 1 or 1 x 3
x
y
2
4
6
8
10
10
2.5
5
7.5
−4
−2
−10
−7.5
−5
−2.5
x
y
10
5
−2
−6
−8
−10
−5
−4
75. x 1 or 0 x 1 77. (e)
79. x 4/3 or x 0 81. x 7 or x 4
83. x 23
or 3 x 23
85. x
1
6
13
or x
1
6
13
87.
3
2
i31
89.
1
3
0
1
3
0
1
i
91.
3
6
, i 93. 2, 1 3
i
95. i, i, 2, 1
97. x
4
2x
3
2x
2
is one possibility
Chapter 4 Test, page 339
1. At t .57735 minutes.
2. (a) Quotient: x
2
x 1 Remainder: 3
(b) Quotient: 2x
3
x
2
1 Remainder: 0
(c) Quotient: x
3
2x
2
x 1 Remainder: 4x
3.
(a) (x 1)
2
2 (b) (x 1)
2
2
(c) (x 1)
2
2 (d) (x 1)
2
2
(e) (x 1)
2
2 (f) (x 1)
2
2
1123
5
10
10
5
x
3 2
y
1123
5
10
10
5
x
3 2
y
1123
5
10
10
5
x
3 2
y
1123
5
10
10
5
x
3 2
y
1123
5
10
10
5
x
3 2
y
1123
5
10
10
5
x
3 2
y
1
3
1000 ANSWERS
(g) (x 1)
2
2 (h) (x 1)
2
2
4. (a) I x(75 20(10 x)) 20x
2
275x (b) $6.88
5. (a) 1, 1,
1
3
(b) 3, 3, 2
, 2
(c) 1, 2 (the other two roots are complex)
6. 10 x 20
7. (a) 3 6i (b) 2 3i, 2 3i (c)
1
2
i,
1
2
i
8. (a)
1
8
(b) Vertical: x 4, x 2 Horizontal: y 0 (the x axis)
(c) (1, 0)
(d)
9. (a) x 2 (b) 1 x 2(c)x
2
3
or x 2
(d) 2 x 3
(e) x
2
3
or x
9
2
10. Asymptote: y 3x
11. (x 1 2i)(x 1 2i)(x 5 i)(x 5 i)x
x
5
12x
4
51x
3
102x
2
130x
Chapter 5
Section 5.1, page 349
1. 1000k
13/4
3. c
42/5
d
10/3
5.
3y
1
2/3
7.
4
a
9
1
b
/
5
2
/2
9. a
x
11. x
7/6
x
11/6
13. x y 15. x y (x y)
3/2
17. (x
1/3
3)(x
1/3
2) 19. (x
1/2
3)(x
1/2
1)
1
5
10
15
1234
2
10
15
5
y
x
510
10
5
2
2
1
1
y
x
1123
5
10
10
5
x
3 2
y
1123
5
10
10
5
x
3 2
y
21. (x
2/5
9)(x
1/5
3)(x
1/5
3)
23. x
1/2
25. a(a b)
1/2
27. 4t
27/10
29. 45
31. 62
33.
2
5
11
35. 2
37. 155
39.
4
b
a
4
41.
43.
a
2
c
3
b
2
45.
3
4
2
47.
33
4
3
49.
2
x
x
4
4
51. 5
3
4
53.
55.
3
2
3
1
57.
59.
61. About 35,863,131 mi 63. About 886,781,537 mi
65. About 5,312,985 67. About 20,003,970
69. (a) Any even power of a real number is never negative.
(b)
3
8
2, but
6
(8)
2
2
71. (a) Since its graph passes the horizontal test, f is one-to-
one and hence has an inverse.
(b) (g f )(x) g( f (x)) ( f (x))
1/5
(x
5
)
1/5
x and
( f g)(x) f (g(x)) (g(x))
5
(x
1/5
)
5
x
(c) Since its graph does not pass the horizontal test, f is not
one-to-one and hence has no inverse.
73. The graph of g is the graph of f (x) x
shifted horizon-
tally 3 units to the left.
75. The graph of k is the graph of f (x) x
shifted horizon-
tally 4 units to the left and vertically 4 units downward.
77. (a)
−30
−10
10
30
2x h
(x h
)
2
1
x
2
1
1
x h
1
x 1
3
9
3
3
1
4
d
5
2c
ANSWERS 1001
(b) If both labor and capital are doubled, output is doubled.
If both are tripled, output is tripled.
79. (a)
(b) If both labor and capital are doubled, output is multiplied
by 2
5/4
. If both are tripled, output is multiplied by 3
5/4
.
Special Topics 5.1.A, page 355
1. x 7 3. x
9
4
5. x 2
7. x 3 9. x 1 or 2 11. x 9
13. x
1
2
15. x
1
2
or x 4
17. x .73
19. x 1.17 or x 2.59 or x 1
21. x 5 or x 3 23. x 6
25. x 3 or 7 27. r 4.658
29. r 8.019 31. b
33. x
1
y
1
2
35. No solution
37. x 36 39. x 1 or 8
41. x 16 43. x 105.236
45. x .283
47. (a) 11.47 ft or 29.91 ft (b) 21 ft
49. Approx. 6.205 mi
51. (a) 1600 feet (b) 9.52 seconds
Section 5.2, page 365
1.
1
3
x
23321
3
6
9
12
18
15
y
a
A
2
1
LCQ L
1/4
C
3/4
10 7 7.65
20 14 15.31
30 21 22.96
40 28 30.61
60 42 45.92
LCQ L
1/2
C
3/4
10 7 13.61
20 14 32.37
30 21 53.73
40 28 76.98
60 42 127.79
3.
5.
7.
9.
11. Shift vertically 5 units down.
13. Stretch away from the x-axis by a factor of 3.
2
x
21431234
1
4
3
5
1
2
3
4
5
6
7
8
9
10
y
2
x
123421
4
6
8
10
2
y
1
x
245 354321
2
3
1
4
5
6
9
8
7
10
y
1
x
1221
3
4
1
2
y
1002 ANSWERS
15. Shift horizontally 2 units left, then shift vertically 5 units
down.
17. Line A: j(x)
Line B: g(x)
Line C: h(x)
Line D: f(x)
19. Neither 21. Even 23. Even
25. 90 27.
2
4
7
29. 1000(2
1.001
2) 1.387
31. 1000(e
1.001
e) 2.720
33.
10
xh
h
10
x
35.
37.
4 x 4, 1 y 10
39. 4 x 4, 1 y 10
41. 10 x 10, 10 y 10
43. 5 x 10, 1 y 6
45. The negative x-axis is an asymptote; (1.443, .531) is a
local minimum.
47. No asymptote; (0, 1) is a local minimum.
49. The x-axis is an asymptote; (0, 1) is a local maximum.
51. (a) 100 (b) 569; 3242 (c) 13 weeks
(d) No, at some point there will not be enough space for
the fruit flies and the rate the fly population grows will
decrease.
53. (a) k 15 (b) About 12.13 psi
(c) About .0167 psi
55. (a) About 74.1 years, about 76.3 years (b) 1930
57. (a) 10 beavers, 149 beavers
(b) After about 9.5 years
59. (a)
(b) C(t) 2
4t
61. (a) f (x) 6(3
x
)orf (x) 18(3
x1
)
(b) 3 (c) No; yes
63. (a)
(b) f(x) .002(2
x
) (c) 2097.15 in. 174.76 ft
(d) 43
2
xh
2
xh
2
x
2
x
h
Time (hours) Number of Cells
01
.25 2
.5 4
.75 8
116
Folds 012 3 4
Thickness .002 .004 .008 .016 .032
65. (a) f (x) 1200(1.04)
x
(b) 1349.84; 1503.57
(c) After approximately 11 years and 1 month
67. (a) 100.4(1.014)
x
(b) 115.3 million (c) 2016
69. (a) 32.44e
.02216x
(b) 40.49 million; 56.45 million (c) 2024
71. About 256; about 654
73. (a) f (x) .75
x
(b) About 8 ft
75. (a) M(x) 5(0.5
x/5730
) (b) 3.08 g; 1.90 g
(c) After about 13,305 years
77. f (x) a
x
for any nonnegative constant a.
79. (a) The graph of f is the mirror image of the graph of g
(b) k(x) f (x); see part (a).
81. (a) Not entirely (b) f
8
(x)
(c) Not at the right side; f
12
(x)
Special Topics 5.2.A, page 374
1. (a) $1469.33
(b) $1485.95
(c) $1489.85
(d) $1491.37
3. 633.39 5. 674.17 7. 819.32
9. 686.84 11. $3325.29 13. $3359.59
15. (a) 7095.36 (b) No
17. Fund C 19. $385.18 21. $1,162,003.14
23. About 5% 25. About 5.92%
27. (a) 9 years (b) 9 years (c) 9 years
(d) Investment amount and doubling time are independent.
29. 9.9 years
31. (a) 907.50
(b) CD, since the CD earns 918.06
(c) 971.27; you would make more money this way.
Section 5.3, page 383
1. 4 3. 2.5
5. 10
3
1000 7. 10
2.88
750
9. e
1.0986
3 11. e
4.6052
.01
13. e
zw
x
2
2y 15. log .01 2
17. log 3 .4771 19. ln 25.79 3.25
21. ln 5.5527 12/7 23. ln w 2/r
25.
43
27. 15 29. .5
31. 931 33. x y 35. x
2
37. f(x) 4e
3.2189x
39. g(x) 16e
3.4177x
41.
e
1
3
x
.0498
x
43. x 1
45. x 0
47. (a) for all x 0.
(b) According to the fourth property of logarithms,
e
ln x
x for every x 0.
ANSWERS 1003
49. They are the same for x 0. If x 0, g(x) does not exist,
while f (x) is symmetric with respect to the y-axis.
51. Stretch vertically by a factor of 2 away from the x-axis.
53. Shift horizontally 4 units to the right.
55. Shift horizontally 3 units to the left, then shift vertically
4 units down.
57.
59.
61.
0 x 20 and 10 y 10; then 0 x 2 and
20 y 20
63. 10 x 10 and 3 y 3
65. 0 x 20 and 6 x 3
67. 10 x 10 and 10 y 10
69. .5493 71. .2386
73. (a)
ln(2 h
h
) ln(2)
(b) .49875415; .499875042; .499987501; .49999875
(c) .24968802; .249968755; .249996875; .24999970
(d) Answers will vary
(e) About 2.5
75. ( f g)(x)
1 e
1
ln(x/1x)
x;
(g f )(x) ln
ln
ln
e
1
x
ln(e
x
) x
1
1
e
x
1
e
e
x
x
1
1
e
x
1
1
1
e
x
1
1
x
1
1
1
x
x
1
1
1
e
ln(x
1
/1x )
x
42
0
86
1
2
3
3
2
1
y
x
42
0
861210 1614 18
1
2
3
3
2
1
y
1
1
1
x
x
77. A 9, B 10
79. About 4392 meters
81. (a) 26.55 billion pounds, 27.72 billion pounds
(b) 2022
83. (a) About 9.9 days
(b) 6986 people
85. (a) No ads: about 120 bikes; $1000: about 299 bikes;
$10,000: about 513 bikes.
(b) $1000: yes, $10,000: no.
(c) $1000: yes, $10,000: yes.
87. n 30 gives an approximation with a maximum error of
.00001 when .7 x .7.
89. (a) Answers will vary.
(b) (28.087, 29.087)
(c) 2529
Section 5.4, page 390
1. ln (x
2
y
3
) 3. log (x 3) 5. ln (x
7
)
7. ln (e 1)
3
9. log (20xy) 11. 2u 5v
13.
1
2
u 2v 15.
2
3
u
1
6
v
17. False; the right side is not defined when x 0, but the left
side is.
19. True by the Power Law
21. False; the graph of the left side differs from the graph of the
right side.
23. False
25. a 10, b 10, among many examples
27. Since v e
ln v
and w e
ln w
, we have
w
v
e
e
l
l
n
n
w
v
e
ln vln w
by the division property of exponents. So raising e to the
exponent (ln v ln w) produces
w
v
. But the exponent to which
e must be raised to produce
w
v
is, by definition, ln
w
v
. There-
fore, ln
w
v
ln v ln w.
29. 4.7 31. 3.176
33. 40 35. 100
37. I must be squared.
39. Following the hint: 10
log c
c because log c is the exponent
to which 10 must be raised to produce c. Taking the natural
logarithm of both sides produces ln (10
log c
) ln (c), but
ln (10
log c
) log c
ln 10 by the power law for logarithms.
Thus, log c
ln 10 ln c, and dividing both sides of this
equation by ln 10 produces the desired result.
41. If c is a positive number, then c can be written in scientific
notation as c b 10
k
, where 1 b 10 and k is an
integer. Then log c log (b 10
k
) log b log (10
k
)
log b k.
1004 ANSWERS
43. (a)
Wayland. It appears to be exponential.
(b)
(c) Wayland. It now appears linear in part b.
Special Topics 5.4.A, page 397
1. x 0124
f (x) log
4
x Not defined 0 .5 1
3. x 1/36 1/6 1 216
h(x) log
6
x 2 10 3
5. x 01/7
7
49
f (x) 2 log
7
x Not defined 214
7. x 2.75 1129
h(x) 3 log
2
(x 3) 63615
9. log .01 2 11. log
3
10
1/3
13. log r 7k 15. log
7
5,764,801 8
17. log
3
(1/9) 2 19. 10
4
10,000
21. 10
2.88
750 23. 5
3
125
25. 2
2
1
4
27. 10
zw
x
2
2y
29.
97
31. x
2
y
2
33. 1/2 35. 6
1
3
2
4
6
y
x
1 5432
1
4
8
12
16
2
6
10
14
18
22
26
20
24
28
32
36
30
34
38
40
x
2 345 6
y
37. b 3 39. b 20 41. 5
43. 3 45. 4 47. log
x
z
2
y
6
3
49. log (x
2
3x) 51. log
2
(5c)
53. log
4
49
1
c
2
55. ln
(x
x
1
2
)
2
57. log
2
(x)
59. ln (e
2
2e 1) ln (e 1)
2
2 ln (e 1)
61. 3.3219 63. .8271 65. 1.1115
67. 1.6199 69. True 71. True
73. False 75. 397
398
77. log
b
u
l
l
o
o
g
g
a
a
u
b
79. log
10
u 2 log
100
u
81. log
b
x
1
2
log
b
v 3
1
2
log
b
v 3 log
b
b log
b
v
1/2
log
b
b
3
log
b
(b
3
v
), so x b
log
b
x
b
log
b
(b
3
v
)
b
3
v
.
83. f (x) g(x) only when x .123, so the statement is false.
85.
87.
(a) It is greater than one google. Since log
5
(10
100
)
143.0677, we have 5
143.0677
10
100
. Since 5
150
5
143.0677
, 5
150
10
100
.
(b) 2
3322
is slightly larger than (10
100
)
10
10
1000
. When we
compare their logarithms we have log 2
3322
3322 log 2 1000.02 and log 10
1000
1000 log 10
1000.
Section 5.5, page 406
1. x 4 3. x 1/9 5. x
1
2
or 3
7. x 2 or 1/2 9. x ln 5/ln 3 1.465
11. x ln 3/ln 1.5 2.7095
13. x
l
l
n
n
3
5
5
2
l
l
n
n
5
3
1.825
15. x
3
ln
ln
2
2
ln
ln
3
3
.1276
17. x (ln 5)/2 .805
19. x (ln 3.5)/1.4 .895
21. x 2 ln(5/2.1)/ln 3 1.579
x
4 26246
h(x) x log x
2
Hole at (0, 0)
(.3679, .3195)
(.3679, .3195)
2
4
6
4
2
y
ANSWERS 1005
23. x 0 or 1
25. x ln 2 .693 or x ln 3 1.099
27. x ln 3 1.099
29. x ln 2/ln 4 1/2orx ln 3/ln 4 .792
31. x ln (t
t
2
1
)
33. (a) If ln u ln v, then u e
ln u
e
ln v
v.
(b) No, if u and v are negative then ln u and ln v are undefined.
35. x 9 37. x 5 39. x 6 41. x 3
43. x
5
2
37
45. x 9/(e 1)
47. x 5 49. x
10001
51. x
e
e
1
1
53. .444 billion years
55. 14.95 days 57. Approximately 3.8132 days
59. About 3689 years old 61. About 2534 years ago
63. About 6.99%
65. (a) About 22.5 years (b) About 22.1 years
67. $3197.05 69. 79.36 years
71. (a) .0146 (b) 2024
73. (a) k 21.459 (b) t .182
75. (a) 20 at the beginning and 2500 3 hours later
(b)
l
l
n
n
2
5
.43
77. (a) 200 people; 2795 people
(b) After 6 weeks
79. (a) 16 years from this year
(b) 24 years from this year
81. (a) c
25
3
0
; k
1
4
ln
2
5
(b) 12.43 weeks
Section 5.6, page 415
1. Linear 3. Logarithmic
5. Exponential, quadratic, cubic, or power
7. Logistic or cubic 9. Quadratic or cubic
11. Ratios are approximately 5.07, 5.06, 5.06, 5.08, and 5.05.
An exponential model might be appropriate.
13. (a) As x increases, e
.0216x
decreases from 1 toward 0.
Thus, (1 56.33e
.0216x
) decreases toward 1, so the
ratio increases toward (but never reaches) 442.1.
(b)
0
500
0 550