43. x ! 0
y !0
y "3#x
3x $ y " 6
y " #3x $ 6
vertices: (0, 0), (0, 3),
(1.5, 1.5), (2, 0)
10.
y
(0, 3)
(1.5, 1.5)
(2, 0)
O (0, 0)
(x, y)
2x $ 4y
(0, 0)
2(0) $ 4(0)
(0, 3)
2(0) $ 4(3)
(1.5, 1.5) 2(1.5) $ 4(1.5)
(2, 0)
2(2) $ 4(0)
max: f(0, 3) % 12
min: f(0, 0) % 0
22x$3 % 33x
log 22x$3 % log 33x
(2x $ 3) log 2 % 3x log 3
2x log 2 $ 3 log 2 % 3x log 3
2x log 2 # 3x log 3 % #3 log 2
x(2 log 2 # 3 log 3) % #3 log 2
x
f(x $ y)
0
12
9
4
1
&&
$2
2
n(log 2 # &1& log 3) % #log 3
2
t#5!
log 12
#0.4771
&&&
1
0.3010 # &&(0.4771)
2
Pages 629–630 Exercises
log 16
13. log5 16 % &
&
log 5
log 8
2.0934
&&
0.9031
!
2a ! 2.3180
a ! 1.159
1.2041
&&
0.6989
! 1.723
log 125
15. log3 125 % &
&
!
log 3
2.0969
&&
0.4771
! 4.395
log 25
17. log12 25 %&
&
log 12
log 2.1
0.9694
&&
0.3222
1.3979
!&
&
1.0792
19.
log 4
2.0969
&&
0.6021
!1.295
9b % 45
log 9b % log 45
b log 9 % log 45
log 45
b%&
&
b!
log 9
1.6532
&&
0.9542
b ! 1.732
y ! 3.483
© Glencoe/McGraw-Hill
n!
ln 2 % ln e0.065t
0.6931 ! 0.065t
10.66 ! t; about 10.7 years
t # 5 ! 3.0087
t ! 8.009
9.
y % log4 125
4y % 125
log 4y % log 125
y ' log 4 % log 125
log 125
y%&
&
y!
#log 3
&&
1
log 2 # && log 3
2
1500
! 1.833
7.
82a % 124
log 82a % log 124
2a log 8 % log 124
log 124
2a % &
&
x ! 2.455
8.
2.1t#5 % 9.32
log 2.1t#5 % log 9.32
(t # 5) log 2.1 % log 9.32
log 9.32
t#5%&
&
n%
n ! #7.638
12. 1500 ( 2 % 3000
A % Pert
3000 % 1500 e(0.065)t
3000
0.065t
&& % e
1
&&
2a !
%
n log 2 % &1&n log 3 # log 3
36 5 .
2. Sample answer: when finding logarithms of
different bases on the calculator
3. Yes; use the change of base formula using a % e,
b % 10, and n % the number.
log 95
log 22
5. log12 95 % &
&
4. log4 22 % &
&
x!
#3(0.3010)
&&&
2(0.3010) # 3(0.4771)
log 2n % log 3 2 n#1
n log 2 % &1&n # 1 log 3
1. Tisha; in exponential equations, the unknown is
an exponent. To solve 36 % x5, Karen must find
log 5
1.7160
&&
0.6990
x!
x ! 1.089
Page 628 Check for Understanding
log 4
#3 log 2
&&
2 log 2 # 3 log 3
#n#
#12#
2n % "3
n
n#2 &2&
2 % (3
)
11.
Solving Exponential
10-6
Equations
! 2.230
6.
5x % 52
log 5x % log 52
x log 5 % log 52
log 52
x%&
&
x%
300
log 82
14. log6 82 % &
&
log 6
1.9138
!&
&
0.7782
! 2.459
log 100
16. log2 100 % &
&
log 2
! &2&
0.3010
! 6.644
log 48
18. log4 48 % &
&
log 4
1.6812
!&
&
0.6021
! 2.792
20.
2x % 30
log 2x % log 30
x log 2 % log 30
log 30
x%&
&
log 2
1.4771
x!&
&
0.3010
x ! 4.907
Algebra 2 Chapter 10
21.
5p % 34
log 5p % log 34
p log 5 % log 34
log 34
p%&
&
27.
log 5
1.5315
p!&
&
log 8
0.6990
2.3010
t!&
&
p ! 2.191
22.
3.1a#3 % 9.42
log 3.1a#3 % log 9.42
(a # 3) log 3.1 % log 9.42
a log 3.1 # 3 log 3.1 % log 9.42
a log 3.1 % log 9.42 $ 3 log 3.1
a%
log 9.42 $ 3 log 3.1
&&&
log 3.1
a!
0.9741 $ 3(0.4914)
&&&
0.4914
0.9031
t ! 2.548
28.
5s$2 % 15.3
log 5s$2 % log 15.3
(s $ 2) log 5 % log 15.3
s log 5 $ 2 log 5 % log 15.3
s log 5 % log 15.3 # 2 log 5
log 15.3 # 2 log 5
s % &&&
log 5
s!
a ! 4.982
23.
6x$2 % 17.2
log 6x$2 % log 17.2
(x $ 2) log 6 % log 17.2
x log 6 $ 2 log 6 % log 17.2
x log 6 % log 17.2 # 2 log 6
x%
log 17.2 # 2 log 6
&&&
log 6
x!
1.2355 # 2(0.7782)
&&&
0.7782
29.
n%
n!
1.1847 # 2(0.6990)
&&&
0.6990
s ! #0.305
9z#4 % 6.28
log 9z#4 % log 6.28
(z # 4) log 9 % log 6.28
z log 9 # 4 log 9 % log 6.28
z log 9 % log 6.28 $ 4 log 9
log 6.28 $ 4 log 9
z % &&&
log 9
z!
x ! #0.412
24.
8.2n#3 % 42.5
log 8.2n#3 % log 42.5
(n # 3) log 8.2 % log 42.5
n log 8.2 # 3 log 8.2 % log 42.5
n log 8.2 % log 42.5 $ 3 log 8.2
30.
log 42.5 $ 3 log 8.2
&&&
log 8.2
0.7980 $ 4(0.9542)
&&&
0.9542
z ! 4.836
7.6a#2 % 41.7
log 7.6a#2 % log 41.7
(a # 2) log 7.6 % log 41.7
a log 7.6 # 2 log 7.6 % log 41.7
a log 7.6 % log 41.7 $ 2 log 7.6
log 41.7 $ 2 log 7.6
a % &&&
log 7.6
a!
1.6284 $ 3(0.9138)
&&&
0.9138
n ! 4.782
25.
t % log8 200
8t % 200
log 8t % log 200
t log 8 % log 200
log 200
t%&
&
31.
x % log5 61.4
5x % 61.4
log 5x % log 61.4
x log 5 % log 61.4
log 61.4
x%&
&
log 5
1.6201 $ 2(0.8808)
&&&
0.8808
a ! 3.839
3.53x$1 % 65.4
log 3.53x$1 % log 65.4
(3x $ 1) log 3.5 % log 65.4
3x log 3.5 $ log 3.5 % log 65.4
3x log 3.5 % log 65.4 # log 3.5
log 65.4 # log 3.5
x % &&&
3 log 3.5
1.7882
x !&
&
0.6990
x!
x ! 2.558
26.
8y#2 % 7.28
log 8y#2 % log 7.28
(y # 2) log 8 % log 7.28
y log 8 # 2 log 8 % log 7.28
y log 8 % log 7.28 $ 2 log 8
y%
log 7.28 $ 2 log 8
&&&
log 8
y!
0.8621 $ 2(0.9031)
&&&
0.9031
32.
20x % 70
2
log 20x % log 70
2
x log 20 % log 70
log 70
x2 % &
&
1.8156 # 0.5441
&&
3(0.5441)
x ! 0.779
2
log 20
1.8451
x2 ! &
&
1.3010
x2 ! 1.4182
x ! ) 1.191
y ! 2.955
© Glencoe/McGraw-Hill
301
Algebra 2 Chapter 10
33.
8x #2 % 32
2
log 8x #2 % log 32
2
(x # 2) log 8 % log 32
x2 log 8 # 2 log 8 % log 32
x2 log 8 % log 32 $ 2 log 8
log 32 $ 2 log 8
x2 % &
&
2
2
x !
39.
log 8
1.5051 $ 2(0.9031)
&&&
0.9031
x!
x2 ! 3.6666
x ! ) 1.915
34.
% 82.9
5.8
2
log 5.8x #3 % log 82.9
(x2 # 3) log 5.8 % log 82.9
2
x log 5.8 # 3 log 5.8 % log 82.9
x2 log 5.8 % log 82.9 $ 3 log 5.8
log 82.9 $ 3 log 5.8
x2 % &&&
40.
log 5.8
!
1.9186 $ 3(0.7634)
&&&
0.7634
y!
41.
log 9 # log 2
a!
42.
log 5 # log 3
0.6990
&&
0.6990 # 0.4771
7t#2
% 5t
t#2
log 7
% log 5t
(t # 2) log 7 % t log 5
t log 7 # 2 log 7 % t log 5
t log 7 # t log 5 % 2 log 7
t(log 7 # log 5) % 2 log 7
2 log 7
t % &&
t!
82y % 524y$3
log 82y % log 524y$3
2y log 8 % (4y $ 3) log 52
2y log 8 % 4y log 52 $ 3 log 52
2y log 8 # 4y log 52 % 3 log 52
y(2 log 8 # 4 log 52) % 3 log 52
3 log 52
y % &&&
y!
2 log 8 # 4 log 52
3(1.7160)
&&&
2(0.9031) # 4(1.7160)
y ! #1.018
43.
log 7 # log 5
2(0.8451)
&&
0.8451 # 0.6990
t ! 11.567
38.
0.3010 $ 2(0.6990)
&&&
5(0.6990) # 2(0.3010)
a ! 0.587
x ! 3.151
37.
55a#2 % 22a$1
log 55a#2 % log 22a$1
(5a # 2) log 5 % (2a $ 1) log 2
5a log 5 # 2 log 5 % 2a log 2 $ log 2
5a log 5 # 2 log 2 % log 2 $ 2 log 5
a(5 log 5 # 2 log 2) % log 2 $ 2 log 5
log 2 $ 2 log 5
a % &&
5 log 5 # 2 log 2
5x#1 % 3x
log 5x#1 % log 3x
(x # 1) log 5 % x log 3
x log 5 # log 5 % x log 3
x log 5 # x log 3 % log 5
x(log 5 # log 3) % log 5
log 5
x % &&
x!
#0.9031
&&&
3(0.6990) # 0.9031
y ! #0.756
9a % 2a
log 9a % log 2a
a log 9 % a log 2
a log 9 # a log 2 % 0
a(log 9 # log 2) % 0
a % &0&
a%0
36.
53y % 8y#1
log 53y % log 8y#1
3y log 5 % (y # 1) log 8
3y log 5 % y log 8 # log 8
3y log 5 # y log 8 % #log 8
y(3 log 5 # log 8) % #log 8
#log 8
y % &&
3 log 5 # log 8
x2 ! 5.5132
x ! ) 2.348
35.
log 8 # log 5
2(0.9031)
&&
0.9031 # 0.6990
x ! 8.849
x2#3
x2
8x#2 % 5x
log 8x#2 % log 5x
(x # 2) log 8 % x log 5
x log 8 # 2 log 8 % x log 5
x log 8 # x log 5 % 2 log 8
x(log 8 # log 5) % 2 log 8
2 log 8
x % &&
16d#4 % 33#d
log 16d#4 % log 33#d
(d # 4) log 16 % (3 # d) log 3
d log 16 # 4 log 16 % 3 log 3 # d log 3
d log 16 $ d log 3 % 3 log 3 $ 4 log 16
d(log 16 $ log 3) % 3 log 3 $ 4 log 16
3 log 3 $ 4 log 16
d % &&&
403x
% 52x$1
log
% log 52x$1
3x log 40 % (2x $ 1) log 5
3x log 4 % 2x log 5 $ log 5
3x log 40 # 2x log 5 % log 5
x(3 log 40 # 2 log 5) % log 5
log 5
x % &&&
403x
x!
3 log 40 # 2 log 5
0.6990
&&&
3(1.6021) # 2(0.6990)
x ! 0.205
log 16 $ log 3
d!
3(0.4771) $ 4(1.2041)
&&&
1.2041 $ 0.4771
d ! 3.716
© Glencoe/McGraw-Hill
302
Algebra 2 Chapter 10
48b. E % 42e0.019t
t
E
0
42
50
108.6
100
280.8
4n % "5
#n#
#&1& 2#
4n % 5(n#2)1 2
&&
log 4n % log 5 2 n#1
44.
n log 4 %
$&2&n # 1% log 5
1
n log 4 % &1&n log 5 # log 5
2
% #log 5
350
300
% #log 5
#log 5
n % &&
1
n!
Events
log 4 # && log 5
2
#0.6990
&&&
1
0.6021 # &&(0.6990)
2
45.
0
x#2
$
log 2
%
% log 8x#2
# &1& log 2 % (x # 2) log 8
3
1
1
&&x log 2 # && log 2 % x log 8 # 2 log 8
3
3
1
1
&&x log 2 # x log 8 % #2 log 8 $ && log
3
3
x &1& log 2 # log 8 % #2 log 8 $ &1& log
3
3
1
#2 log 8 $ && log 2
3
1
x % &&&
&& log 2 # log 8
3
$
%
x!
46.
2
5x
2
1
#2(0.9031) $ &&(0.3010)
3
&&&
1
#
&&(0.3010) 0.9031
3
log n
x % &b&
logb a
47. 2500 ' 2 % 5000
P(t) % Po ekt
5000 % 2500 e(0.03)t
2 % e0.03t
ln 2 % ln e0.03t
0.6931 ! 0.03t
23 ! t; about 23 years
48a.
E % 42ekt
350 % 42e0.019t
350
0.019t
&& % e
42
350
&&
42
50
75
Years
100
125
t
x
16
&&
x
$
30
%
$ &1& % 5x(1)
5
9.5
38
53. (x $ 2)(x $ 9) * 0
x $ 2 * 0 and x $ 9 * 0 or x $ 2 + 0 and x $ 9 + 0
x * #2
x * #9
x + #2
x +#9
x * #2
or
x + #9
{x| x * #2 or x + #9}
54. Let x % width, then length % &1&(6 # 2x) or 3 # x
2
A % x(3 # x)
2
A % 3x # x
x A
y
0 0
(1.5, 2.25)
1 2
2 2
3 0
O
x
Maximum area is
at x % 1.5.
1.5 feet by 1.5 feet
% ln e0.019t
ln 350 # ln 42 % 0.019t
5.8579 # 3.7377 ! 0.019t
2.1202 ! 0.019t
111.6 ! t
111.6 years or by the 2008 Summer Olympics
© Glencoe/McGraw-Hill
25
80 $ x % 5x
#4x % #80
x % 20; 20 days
51.
b4 # 5b2 $ 4 % 0
(b2)2 # 5(b2) $ 4 % 0
(b2 # 4)(b2 # 1) % 0
(b # 2)(b $ 2)(b # 1)(b $ 1) % 0
b # 2 % 0 or b $ 2 % 0 or b # 1 % 0 or b $ 1 % 0
b%2
b % #2
b%1
b % #1
52.
4y2 # x2 # 24y $ 6x % 11
4y2 # 24y # x2 $ 6x % 11
4(y2 # 6y) # (x2 # 6x) % 11
2
4(y # 6y $ 9) # (x2 # 6x $ 9) % 11 $ 4(9) # 9
4(y # 3)2 # (x # 3)2 % 38
(y # 3)2
(x # 3)2
&& # && % 1; hyperbola
x ! 2.125
x % loga n
ax % aloga n
ax % n
logb ax % logb n
x logb a % logb n
ln
0
Sample answer: Both show that the number of
events is increasing more and more at each
Olympics.
49. 0.0023
50. Let x % the time it would take the painter to do
the job alone.
16
6
&& $ && % 1
1
&&
1
&&x
3
E ! 42 e0.019t
50
2(x#1)13 % 8x#2
1
&&x#&&
3
3
200
100
"2### % 8
x#1
250
150
n ! #2.767
3
E
400
n log 4 # &1&n log 5
2
n(log 4 # &1& log 5)
2
303
Algebra 2 Chapter 10
55. 2x $ 5yi % 4 $ 15i
2x % 4
5yi % 15i
x%2
y%3
&
56. M#1 % &1&
#1 #5
#6 # 15 #3 6
% &1&
&
#1 #5
#21 #3 6
4.
'
'
log 0.8
3.7 ! n; about 3.7 years
y % n ekt
1 % 2 ek(1620)
0.5 % e1620k
ln 0.5 % ln e1620k
#0.6931 ! 1620k
#0.00043 ! k; about #0.00043
5b. y % n e#0.00043t
5c. y % n e#0.00043t
% 20e#0.00043(5000)
% 20e#2.15
! 2.33; about 2.33 grams
5d.
y % n e#0.00043t
1 % 4e#0.00043t
0.25 % e#0.00043t
ln 0.25 % ln e#0.00043t
#1.3863 ! #0.00043t
3224 ! t; in about 3224 years
5e. Never; the amount left will always be half of the
amount that existed 1620 years ago.
& 63 #15 ' & xy ' % & 87 '
6 5
x
8
#1 #5
#1 #5
#&1&&
'
'
% #&1&&
'
21 #3 6 ' & 3 #1 ' & y '
21 #3 6 ' & 7 '
#8 #35
& xy ' % #&21&1 & #24
'
42
5a.
& xy ' % #&21&1 & #4318 '
&y' %
x
57. perpendicular slope %
y # y1 % m(x # x1)
y # 6 % #&3&(x # 4)
#&3&
2
43
&&
21
& '$
#&6&
;
7
43
&&,
21
%
#&6&
7
2
y # 6 % #&3&x $ 6
2
y % #&3&x $ 12
3
&&x
2
2
$ y % 12
3x $ 2y % 24
Pages 634–636 Exercises
P(1 $ r)n
75,000(1 $ 0.06)5
75,000(1.06)5
100,367; about $100,367
y % n e#0.0856t
1 % 2 e#0.0856t
0.5 % e#0.0856t
ln 0.5 % ln e#0.0856t
#0.6931 ! #0.0856t
8.1 ! t; about 8.1 days
8a. Let x % the distance from the left side and
y % the frequency. Since 535 Kilohertz is 0
centimeters from the left side and 1705
Kilohertz is 15 centimeters from the left side,
the ordered pairs (0, 535) and (15, 1705) are
solutions to the function y % abx.
y % abx
y % abx
0
535 % ab
1705 % 535b15
535 % a
log 1705 % log (535b15)
log 1705 % log 535 $ 15 log b
log 1705 # log 535 % 15 log b
10-7 Growth and Decay
6. Vn %
Vn %
Vn %
Vn !
7.
Pages 633–634 Check for Understanding
1. Sample answer: The constant is positive when
growth is depicted such as in the case of bacteria
growth. The constant is negative when decay is
depicted such as in radioactive decay. If k is zero,
then the population is not growing or decaying
and the function is a constant function.
2. Example 1 is dealing with e, and Examples 2 and
3 are dealing with base 10 numbers; yes; yes; You
can take the logarithm of each side of an equation
as long as you use the same base on both sides.
Therefore, you can use natural logarithms or
common logarithms in any problem.
3a. y % 88(1.0137)x
% 88(1.0137)3.5
! 92.3; about 92.3 megahertz
3b.
y % 88(1.0137)x
100 % 88(1.0137)x
log 100 % log 88 $ x log 1.0137
log 100 # log 88 % x log 1.0137
log 100 # log 88
&&
log 1.0137
log 1705 # log 535
&&&
15
% log b
0.0336 ! log b
antilog 0.0336 ! antilog (log b)
1.0803 ! b
about y % 535(1.0803)x
8b. y % 535(1.0803)x
% 535(1.0803)4.5
! 757; about 757 Kilohertz
%x
9.4 ! x; about 9.4 cm from the left
side
© Glencoe/McGraw-Hill
Vn % P(1 $ r)n
2000 % 4600(1 # 0.20)n
0.4348 ! 0.8n
log 0.4348 % log 0.8n
log 0.4348 % n log 0.8
log 0.4348
&& % n
304
Algebra 2 Chapter 10