### [] You should know the definition of the natural logarithmic function

```PART I: Solutions to Odd-Numbered Exercises and Practice Tests
mCONTINUED--.
[] You should know the definition of the natural logarithmic function.
logex = lnx, x>O
[] You should know the properties of the natural logarithmic function.
(a) In 1 = 0 since e° = 1.
(b) In e = 1 since e1 = e.
(c) In ex = x since ex = ex.
(d) If In x = In y, then x = y.
[] You should be able to graph logarithmic functions.
Solutions to Odd-Numbered Exercises
1. log,, 64 = 3 ==~ 43 = 64
3. 1og7~=-2 ==~ 7-2=~9
5. log324=’~ ==~ 322/5 =4
7. lnl=O =, e°=l
9. 53= 125 =:~ log5125=3
11. 811/’t=3 ~ logs13=¼
1 ==~ log6 3~ = -2
13. 6-2 -- 36
15. e3 = 20.0855... =:~ In 20.0855... = 3
17. e2"6 = 13.463... ~ In 13.463... = 2.6
19. log2 16 = log2 24 = 4
21. log16(¼) = 1og161 --Io9164 = 0- Io916161/2 = --½23. loglo 0.01 = loglo 10-2 = -2
25. log7x = 10979
x=9
27. In es = x
29. log6 62 = x
8 ¯ In e = x
2log6 6 = x
8=X
2=X
31. loglo 345 ~ 2.538
33. log~o(~) ~--0,097
35. In(4 + ~"~)~ 1.746
37. ln.v/~ ~- 1.869
39. 6 lOglo 14.8 ~ 7.022
41o 12 In 6.4 ~ 22.276
151
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
45. f(x) = ex, g(x) = In x
43. f(x) = 3x, g(x) = log3 x
2
2
fand g are inverses. Their graphs are
reflected about the line y = x.
47. f(x) = log3 x + 2
f and g are inverses. Their graphs are
reflected about the line y = x.
49. f(x) = -log3(x + 2)
Asymptote: x = -2
Point on graph: (- 1, 0)
Matches graph (d).
Asymptote: x = 0
Point on graph: (1, 2)
Matches graph (c).
51. f(x) = log3(1 ’-- x)
Asymptote: x = 1
Point on graph: (0, 0)
Matches graph (b).
53. f(x) = log4 x
Domain: x > 0 =~ The domain is (0, ~x~).
Vertical asymptote: x = 0
x-intercept: (1, 0)
y=log4x ==~ 4y=x
x
Y
1
1
4
2
-1
0
1
1_
55. h(x) = log4(x- 3)
Domain: x - 3 > 0 or (3, ~)
Vertical asymptote: x = 3
Intercept: (4, 0)
y
y
6
4
2.
57, y = -log3 x + 2
Domain: (0,
Vertical asymptote: x = 0
x-intercept: -log3 x + 2 = 0
2 = log3 x
32 =X
9=x
The x-intercept is (9, 0).
y = -loga x + 2
log3 x = 2 - y =:~ 32.-y = x
y
6
4
2
x
27
9
3
1
1
y
-1
0
1
2
3
4
6 8 10 12
bered Exercises and Practice Tests
f(x) =6+log6(x-3)
Domain: (3, ~x~)
Vertical asymptote: x = 3
y
9:
7
x-intercept: log6(x - 3) = -6
5-
6-6=x- 3
X= 3 +6-6~.3
X
4 9
Y 6
7 5
Domain: -_ > 0 =e, x > 0
5
The domain is (0, ~).
x
Vertical asymptote: ~ = 0 ==~ x = 0
The vertical asymptote is the y-axis.
x-intercept: loglo = 0
- = 10o
5
x
x
-=1 ~ x=5
5
The x-intercept is (5, 0).
63. f(x) = ln(x- 2)
Domain: x-2>0 ==~ x>2
The domain is (2, ~).
Vertical asymptote: x - 2 = 0 ==~ x = 2
x-intercept: 0 = ln(x - 2)
eO=x-2
3=x
The x-intercept is (3, 0).
6
4.
32I.~,~-
-1
456789
153
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
g(x)= ln(-x)
Domain: -x > 0 =e, x < 0
The domain is (-c~, 0).
Vertical asymptote: -x = 0 ==~ x = 0
x-intercept: 0 = ln(-x)
e0 m --x
t
-lO
The x-intercept is (- 1, 0).
67. f(x) = :
x
69. h(x) = 4xlnx
(a)
(a)
Iiiii
-/
tii
"11
.(b) Domain: (0, o c)
(c) Increasing on (2, oo)
Decreasing on (0, 2)
(b) Domain: (0, ~)
(c) Increasing on (0.368, ~)
(d) Relative minimum: (2, 1.693)
(d) Relative minimum: (0.368, - 1.472)
71. t =
Decreasing on (0, 0.368)
10 In 2
~ 23.68 years
In 67 - In 50
73. (a) f(0) = 80 - 17 loglo(0 + 1) = 80
(b) f(4) = 80- 17 loglo(4 + 1)= 68.1
(c) f(10) = 80 - 17 log10(10 + 1) ~- 62.3
(d) ,oo
O~
~ 12
0
75. (a)
r
0.005
0.010
0.015
t
138.6 yr
69.3 yr
46.2 yr
r
0.020
0.025
0.030
t
34.7 yr
27.7 yr
23.1 yr
11o
(b) T > 300°F when p > 67.3 pounds per square
inch
[The graph of T and y = 300 intersect at
p = 67.3.]
(c) T(74)= 306.48°F
The doubling time decreases as r increases.
154
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
(a) I= 1:/3= 101ogl0(l~_t~)= 10. loglo(1012) = 10(12) = 120decibels
’
. / lO-~
(b) I = 10-2:/3 = 101oglo~) = I0 loglo(lOI°) = I0(I0)= I00 decibels
(c) No, this is a logarithmic scale.
81. y = 80.4 - 11 lnx, 100_<x< 1500
450 cubic ft per minute
(a)
= 15 cubic feet per minute per child
30 children
(b) From the graph, for y = 15 you get x ~ 382 cubic feet.
(c) If ceiling height is 30, then 382 square feet of floor space is needed.
. [ 1659.24
~
83, t = 16.625 Jn[.165~-~~ : :/50] ~ 10 years
85. Total amount = (1659.24)(10)(12) ~ \$199,108.80
Interest = 199,108.80- 150,000 = \$49,108.80
87. True. 1og3(27) = log3 33 = 3
91. (a)
x
1
0
~(x)
5
10
89.
lO~
0.322 0.230 0.046
lO~
106
0.00092
0.0000138
4
(b) As x increases without bound, f(x) approaches O.
(c) o.~
93. Vertical asymptote: x = 0
No horizontal asymptote
95. Vertical asymptote: x = 7
No horizontal asymptote
97. e7/2 ~ 33.115
99. 6e-s ~- 0.002
```