Transcript No Slide Title

Active Learning Lecture Slides
For use with Classroom Response Systems
Chapter 6
Exponential
and
Logarithmic
Functions
© 2009 Pearson Education, Inc.
All rights reserved.
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 1
Given f x   x  5 and g x   8x  9 find
 f og x .
a.
2 2x  1
b.
2 2x  1
c.
8 x5 9
d.
8 x4
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 2
Given f x   x  5 and g x   8x  9 find
 f og x .
a.
2 2x  1
b.
2 2x  1
c.
8 x5 9
d.
8 x4
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 3
1
Given f x   x  1 and g x  
find
x7
the domain of  f og x .
a.
x x  1, x  7
b.
x 7  x  8
c.
x x  7, x  1
d.
x x is any real number
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 4
1
Given f x   x  1 and g x  
find
x7
the domain of  f og x .
a.
x x  1, x  7
b.
x 7  x  8
c.
x x  7, x  1
d.
x x is any real number
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 5
Find the inverse of the function and state its
domain and range. 4,5 , 2,6 , 0, 7 , 2,8 
a.
6, 5 , 5, 0 , 4, 2 , 6, 7 
D  6, 5, 4; R  5, 0, 2, 7
b.
5, 4 , 6, 2 , 7, 0 , 8, 2 
D  5, 6, 7, 8; R  4, 2, 0, 2
c.
5, 4 , 6, 2 , 7, 0 , 8, 2 
D  5, 6, 7, 8; R  4, 2, 0, 2
d.
6, 5 , 8, 0 , 4, 0 , 6, 7 
D  6, 8, 4; R  5, 0, 7
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 6
Find the inverse of the function and state its
domain and range. 4,5 , 2,6 , 0, 7 , 2,8 
a.
6, 5 , 5, 0 , 4, 2 , 6, 7 
D  6, 5, 4; R  5, 0, 2, 7
b.
5, 4 , 6, 2 , 7, 0 , 8, 2 
D  5, 6, 7, 8; R  4, 2, 0, 2
c.
5, 4 , 6, 2 , 7, 0 , 8, 2 
D  5, 6, 7, 8; R  4, 2, 0, 2
d.
6, 5 , 8, 0 , 4, 0 , 6, 7 
D  6, 8, 4; R  5, 0, 7
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 7
3
f
x

x
 5 as a solid
Graph the function  
curve and its inverse as a dashed curve.
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 8
3
f
x

x
 5 as a solid
Graph the function  
curve and its inverse as a dashed curve.
a.
b.
c.
d.
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Slide 6 - 9
2x  8
The function f x  
is one-to-one. Find
4
x

1
its inverse.
x8
a. f x  
4x  2
1
4x  2
c. f x  
x8
1
Copyright © 2009 Pearson Education, Inc.
b.
2x  2
f x  
4x  1
d.
2x  8
f x  
4x  1
1
1
Slide 6 - 10
2x  8
The function f x  
is one-to-one. Find
4
x

1
its inverse.
x8
a. f x  
4x  2
1
4x  2
c. f x  
x8
1
Copyright © 2009 Pearson Education, Inc.
b.
2x  2
f x  
4x  1
d.
2x  8
f x  
4x  1
1
1
Slide 6 - 11
Graph f x   2
x
 3.
a.
b.
c.
d.
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Slide 6 - 12
Graph f x   2
x
 3.
a.
b.
c.
d.
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Slide 6 - 13
Solve the equation.
a.
13 
 
3
b.
 13 
 
 3
c.
3
 
13 
d.
 3
 
 13 
Copyright © 2009 Pearson Education, Inc.
8
4 x1
 32
5x
Slide 6 - 14
Solve the equation.
a.
13 
 
3
b.
 13 
 
 3
c.
3
 
13 
d.
 3
 
 13 
Copyright © 2009 Pearson Education, Inc.
8
4 x1
 32
5x
Slide 6 - 15
Change 5  x to an equivalent expression
involving logarithms.
3
a.
log5 3  x
b.
log x 5  3
c.
log 3 x  5
d.
log5 x  3
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 16
Change 5  x to an equivalent expression
involving logarithms.
3
a.
log5 3  x
b.
log x 5  3
c.
log 3 x  5
d.
log5 x  3
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 17
1
.
Find the exact value of log 4
64
a.
1

3
b.
3
c.
1
3
d.
3
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Slide 6 - 18
1
.
Find the exact value of log 4
64
a.
1

3
b.
3
c.
1
3
d.
3
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Slide 6 - 19


2
log
x
 x 1
Solve the equation.
72
a.
1, 72 
b.
8, 9 
c.
8, 9 
d.
8, 9 
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 20


2
log
x
 x 1
Solve the equation.
72
a.
1, 72 
b.
8, 9 
c.
8, 9 
d.
8, 9 
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 21
Suppose that ln 2 = a and ln 5 = b. Use the
properties of logarithms to write ln 10 in terms
of a and b.
a.
ab
b.
ab
c.
ab
d.
ln a  lnb
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 22
Suppose that ln 2 = a and ln 5 = b. Use the
properties of logarithms to write ln 10 in terms
of a and b.
a.
ab
b.
ab
c.
ab
d.
ln a  lnb
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 23
3
5
Write log19 2 as the sum and/or difference of
q p
logarithms. Express powers as factors.
a.
b.
c.
d.
3log19 5  2 log19 q  log19 3
1
log19 5  2 log19 q  log19 p
3
1
log19 5  2 log19 q  2 log19 p
3
log19 5  log19 q  log19 p
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 24
3
5
Write log19 2 as the sum and/or difference of
q p
logarithms. Express powers as factors.
a.
b.
c.
d.
3log19 5  2 log19 q  log19 3
1
log19 5  2 log19 q  log19 p
3
1
log19 5  2 log19 q  2 log19 p
3
log19 5  log19 q  log19 p
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 25
Use the Change-of-Base Formula and a
calculator to evaluate log 7 42.60.
a.
6.086
b.
1.629
c.
0.519
d.
1.928
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 26
Use the Change-of-Base Formula and a
calculator to evaluate log 7 42.60.
a.
6.086
b.
1.629
c.
0.519
d.
1.928
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 27
Solve the equation log 3 x  log 3 x  24   4.
a.

b.
3, 27
c.
53
d.
27
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Slide 6 - 28
Solve the equation log 3 x  log 3 x  24   4.
a.

b.
3, 27
c.
53
d.
27
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Slide 6 - 29
Solve the equation 32 x  3x  6  0.
a.
 ln 2 


 ln 3 
b.
 ln 3 


 ln 2 
c.
 ln 2 


 ln 6 
d.
ln 6
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Slide 6 - 30
Solve the equation 32 x  3x  6  0.
a.
 ln 2 


 ln 3 
b.
 ln 3 


 ln 2 
c.
 ln 2 


 ln 6 
d.
ln 6
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 31
Use a graphing utility to solve the equation
x
e  ln x  3.
a.
1.14
b.
2.17
c.
1.27
d.
0.57
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 32
Use a graphing utility to solve the equation
x
e  ln x  3.
a.
1.14
b.
2.17
c.
1.27
d.
0.57
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 33
Austin invested $12,000 in an account at 11%
compounded quarterly. Find the amount in
Austin’s account after a period of 6 years.
a. $11,011.51
b. $23,011.51
c. $22,444.97
d. $22,395.63
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 34
Austin invested $12,000 in an account at 11%
compounded quarterly. Find the amount in
Austin’s account after a period of 6 years.
a. $11,011.51
b. $23,011.51
c. $22,444.97
d. $22,395.63
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 35
A local bank advertises that it pays interest on
savings accounts at the rate of 3% compounded
monthly. Find the effective rate.
a. 3.44%
b. 3.40%
c. 3.04%
d. 36%
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 36
A local bank advertises that it pays interest on
savings accounts at the rate of 3% compounded
monthly. Find the effective rate.
a. 3.44%
b. 3.40%
c. 3.04%
d. 36%
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 37
What principal invested 8% compounded
continuously for 4 years will yield $1190?
a. $627.48
b. $1638.78
c. $1188.62
d. $864.12
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 38
What principal invested 8% compounded
continuously for 4 years will yield $1190?
a. $627.48
b. $1638.78
c. $1188.62
d. $864.12
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 39
Gillian has $10,000 to invest in a mutual fund.
Using average annual rate of return for the past
five years of 12.25%, determine how long it will
take for her investment to double.
a. 6 years
b. 12 years
c. 3 years
d. 4 years
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 40
Gillian has $10,000 to invest in a mutual fund.
Using average annual rate of return for the past
five years of 12.25%, determine how long it will
take for her investment to double.
a. 6 years
b. 12 years
c. 3 years
d. 4 years
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 41
The size P of a small herbivore population at time
t (in years) obeys the function P(t) = 600e0.14t if
they have enough food and the predator
population stays constant. After how many years
will the population reach 1200?
a. 4.95 years
b. 45.69 years
c. 12.09 years
d. 12.8 years
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 42
The size P of a small herbivore population at time
t (in years) obeys the function P(t) = 600e0.14t if
they have enough food and the predator
population stays constant. After how many years
will the population reach 1200?
a. 4.95 years
b. 45.69 years
c. 12.09 years
d. 12.8 years
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 43
The half-life of silicon-32 is 710 years. If 20
grams is present now, how much will be present in
400 years?
a. 13.534
b. 19.234
c. 0.403
d. 0
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 44
The half-life of silicon-32 is 710 years. If 20
grams is present now, how much will be present in
400 years?
a. 13.534
b. 19.234
c. 0.403
d. 0
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 45
A thermometer reading 93ºF is placed inside a
cold storage room with a constant temperature of
36ºF. If the thermometer reads 88ºF in 14 minutes,
how long before it reaches 54ºF? Use Newton’s
Law of Cooling: U = T + (U0 – T)ekt.
a. –37 min
b. 176 min
c. –2 min
d. 40 min
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 46
A thermometer reading 93ºF is placed inside a
cold storage room with a constant temperature of
36ºF. If the thermometer reads 88ºF in 14 minutes,
how long before it reaches 54ºF? Use Newton’s
Law of Cooling: U = T + (U0 – T)ekt.
a. –37 min
b. 176 min
c. –2 min
d. 40 min
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 47
360
The logistic growth function f t  
0.27t
1  11e
describes the population of a species of butterflies
t months after they are introduced to a nonthreatening habitat. How many butterflies are
expected in the habitat after 20 months?
a. 600 butterflies
b. 343 butterflies
c. 360 butterflies
d. 7200 butterflies
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 48
360
The logistic growth function f t  
0.27t
1  11e
describes the population of a species of butterflies
t months after they are introduced to a nonthreatening habitat. How many butterflies are
expected in the habitat after 20 months?
a. 600 butterflies
b. 343 butterflies
c. 360 butterflies
d. 7200 butterflies
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 49
A mechanic is testing the cooling system of a boat
engine. He measures the engine’s temperature
over time. Use a graphing utility to build a
logistic model from the data.
Time, min
Temperature, ºF
5
10 15 20 25
100 180 270 300 305
314.79
a. y 
0.246 x
1  7.86e
314.79
b. y 
1.22 x
1  7.86e
311.63
c. y 
1  8.1e0.253x
306.53
d. y 
1  7.92e0.254 x
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 50
A mechanic is testing the cooling system of a boat
engine. He measures the engine’s temperature
over time. Use a graphing utility to build a
logistic model from the data.
Time, min
Temperature, ºF
5
10 15 20 25
100 180 270 300 305
314.79
a. y 
0.246 x
1  7.86e
314.79
b. y 
1.22 x
1  7.86e
311.63
c. y 
1  8.1e0.253x
306.53
d. y 
1  7.92e0.254 x
Copyright © 2009 Pearson Education, Inc.
Slide 6 - 51