Chapter 12 Exponential and Logarithmic Functions
71.
33.4 1.66 t
ft
421 33.4 1.66
t
72. a.
11ln 49
51 11ln 51 49
92
Wx x
W
73.
4
4
4
0.065
20,000 12,500 1 4
20, 000 12,500 1 0.01625
t
t
t
74.
0.075
0.075
3 50,000 50,000
350,000
t
t
e
e
ln 3 0.220
5
r
The money will triple in 5 years if the interest rate is
0.036
10
0.036
35.3
t
k
k
Ae
b. Note that 2015 is 15 years after 2000, find A for
15.t
Chapter 12 Test
c.
0.036
70 35.3 t
e
0.036
70
35.3
t
e
77. Find k:
0
140
0
0
2
kt
k
AAe
AAe
78. a. Scatter plot:
79. a. Scatter plot:
80. a. Scatter plot:
Chapter 12 Test
1.
1
2
2
x
x
fx
gx
Chapter 12 Exponential and Logarithmic Functions
2. Semiannual Compounding:
210
20
0.065
3000 1 2
3000 1.0325 5687.51
A
3.
5
3
log 125 3
5125
5.
3
3
log
x
fx
gx x
6. Since
ln ,
x
ex
5
ln 5 .
x
ex
7.
log 1
b
because
1
bb
.
9.
5
log 7fx x
70
7
x
x
The domain of f is
7, .
12
10log10 10 12 120
The sound has a loudness of 120 decibels.
1
3
33
81
1
log 4 log 4
3
xx
13.
62
62
6 log 2 log log log
log( )
xyxy
xy
The solution set is
6.
17.
51.4
x
Chapter 12 Test
18.
0.005
0.005
400 1600
1600
x
x
e
e
19.
25
1
2
1
log 2
25 25 5
x
x
The solution set is {5}.
21.
2ln(3 ) 8
8
ln(3 ) 2
x
x
22.
log log 15 2
log 15 2
xx
xx
equation. The solution set is {5}
23.
ln 4 ln( 1) ln 6xx
4
ln ln 6
1
46
x
x
x
24. a.
0.004
0.004 0
82.3
82.3 82.3
x
Ae
Ae
In 2010, the population of Germany was 82.3
Chapter 12 Exponential and Logarithmic Functions
25.
4
4
0.05
8000 4000 1 4
8000 1 0.0125
t
t
26.
10
10
21
r
r
e
27. In 2010, t = 0 and A
0
= 4121
In 2050, t = 2050 – 2010 = 40 and A = 5231.
40
40
5231 4121
5231
4121
k
k
e
e
28.
0.000121
0
0.000121
0.000121
5100
5
t
t
t
AAe
e
30. Plot the ordered pairs.
Cumulative Review
31. Plot the ordered pairs.
32. Plot the ordered pairs.
33.
ln 0.38
0.968
96 0.38
96
96
x
x
x
y
ye
ye
Cumulative Review Exercises
(Chapters 1 – 12)
1.
84 5 7
xx
2.
54 22
38 18
xy
xy
Multiply the first equation by 2 and solve by
addition.
10 8 44
xy
412
3
y
y
The solution set is
2, 3 .
3.
324 6
xyz
Multiply the second equation by 3 and add to the
third equation to eliminate y.
21 3 9 69
23 7
23 10 76
xyz
xyz
xz
The system of two variables in two equations is:
11 10 52
23 10 76
xz
xz
Chapter 12 Exponential and Logarithmic Functions
Back-substitute 2 for x to find z.
11 2 10 52
z
4.
13
13or 13
x
xx
5.
442xx
42 4
xx
6.
40and 3 6
42
xx
xx
For a value to be in the solution set, it must satisfy
7.
2
2
232
2320
xx
xx
The solutions are
37
44
i
, and the solution set is
Find the x–intercept by setting y = 0 and solving.
31550x
315
x
Cumulative Review
9.
23 6xy
First, find the intercepts to the equation
23 6xy
.
This is a false statement. This means that the origin
will not fall in the shaded half-plane.
10.
11
2
fx x
1; intercept 1
2
my
11.
2
68fx x x
Since
1a
is positive, the parabola opens upward.
The x–coordinate of the vertex of the parabola is
42
xx
The x–intercepts are –4 and –2. Set x = 0 and solve
for y to obtain the y–intercept.
2
0608
008
y
y
12.
2
34fx x
Since
1a
is positive, the parabola opens upward.
Chapter 12 Exponential and Logarithmic Functions
Set x = 0 and solve for y to obtain the y–intercept.
2
03 4
y
13.
cd
Acd
Ac d cd
Ac Ad cd
Ac cd Ad
cA d Ad
15.
2
2
() 2
3152
317
gfx fx
xx
xx
17.
73
73
fx x
yx
18.
2
2
32
x
fx xx
To find the domain, find all values of x for which
28x
is greater than zero.
280
28
4
x
x
x
The domain of f is
4,
.
be
2
Using the point,
2, 4 ,
and the slope,
1,
2
we can
write the equation in point-slope form.
11
yy mxx
2
Solve for y to obtain slope-intercept form.
1
Cumulative Review
21.
37 72 9 9
42
5
333
15
xy yy y y
xxx
xy
23.
2
32
32
51
515 24 09
5
xx
xx xx
xx
24.
10 10
3
32
2
3
8
3
62 2 2
33
32 32
2
2
16
82 2 2
xy xy
xy
xy
y
yy y y
26.
43
3
4832
48 4
xxx
xx x
22
26923
xxyy xy
28.
1
22
1
2ln ln ln ln
2
xyxy
Let 2x + 4 = the length of the carpet.
2
2
2
2448
2448
24480
2240
640
xx
xx
xx
xx
xx
Chapter 12 Exponential and Logarithmic Functions
30. Let xtime it takes when working together.
Part done Time Part done
in Working in
1 hour Together hours
1
You 22
x
x
x
If you and your sister work together, it will take
6
5
hours, or 1 hour and 12 minutes, to clean the house.
31. Let xthe rate of the current.
time
distance rate
with the 20
20 15
current 15
d
dr
tr
xx
32.
(10)
10
10
18,000 6000
3
ln 3 ln
ln 3 10
rt
r
r
r
APe
e
e
e
r