slide 6 - 1 copyright © 2009 pearson education, inc. active learning lecture slides for use with...
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Slide 6 - 1Copyright © 2009 Pearson Education, Inc.
Active Learning Lecture SlidesFor use with Classroom Response Systems
© 2009 Pearson Education, Inc.All rights reserved.
Chapter 6Exponential
and Logarithmic Functions
Slide 6 - 2Copyright © 2009 Pearson Education, Inc.
Given
a.
b.
c.
d.
f x x 5 findg x 8x 9and
f og x .
8 x 4
8 x 5 9
2 2x 1
2 2x 1
Slide 6 - 3Copyright © 2009 Pearson Education, Inc.
Given
a.
b.
c.
d.
f x x 5 findg x 8x 9and
f og x .
8 x 4
8 x 5 9
2 2x 1
2 2x 1
Slide 6 - 4Copyright © 2009 Pearson Education, Inc.
Given
a.
b.
c.
d.
f x x 1 findg x 1
x 7and
f og x .
x x is any real number
x x 7, x 1
x 7 x 8
x x 1, x 7
the domain of
Slide 6 - 5Copyright © 2009 Pearson Education, Inc.
Given
a.
b.
c.
d.
f x x 1 findg x 1
x 7and
f og x .
x x is any real number
x x 7, x 1
x 7 x 8
x x 1, x 7
the domain of
Slide 6 - 6Copyright © 2009 Pearson Education, Inc.
Find the inverse of the function and state its domain and range.
a.
b.
c.
d.
5, 4 , 6,2 , 7,0 , 8, 2 D 5,6, 7,8 ; R 4,2,0, 2
5, 4 , 6,2 , 7,0 , 8, 2 D 5, 6, 7, 8 ; R 4,2,0, 2
6,5 , 5,0 , 4,2 , 6, 7 D 6,5, 4 ; R 5,0,2, 7
6,5 , 8,0 , 4,0 , 6, 7 D 6,8, 4 ; R 5,0, 7
4,5 , 2,6 , 0, 7 , 2,8
Slide 6 - 7Copyright © 2009 Pearson Education, Inc.
Find the inverse of the function and state its domain and range.
a.
b.
c.
d.
5, 4 , 6,2 , 7,0 , 8, 2 D 5,6, 7,8 ; R 4,2,0, 2
5, 4 , 6,2 , 7,0 , 8, 2 D 5, 6, 7, 8 ; R 4,2,0, 2
6,5 , 5,0 , 4,2 , 6, 7 D 6,5, 4 ; R 5,0,2, 7
6,5 , 8,0 , 4,0 , 6, 7 D 6,8, 4 ; R 5,0, 7
4,5 , 2,6 , 0, 7 , 2,8
Slide 6 - 8Copyright © 2009 Pearson Education, Inc.
Graph the function as a solid curve and its inverse as a dashed curve.
a. b.
c. d.
f x x3 5
Slide 6 - 9Copyright © 2009 Pearson Education, Inc.
Graph the function as a solid curve and its inverse as a dashed curve.
a. b.
c. d.
f x x3 5
Slide 6 - 10Copyright © 2009 Pearson Education, Inc.
The function is one-to-one. Find its inverse.
a.
c. f 1 x 2x 8
4x 1f 1 x 4x 2
x 8
f 1 x 2x 2
4x 1f 1 x x 8
4x 2
f x 2x 8
4x 1
b.
d.
Slide 6 - 11Copyright © 2009 Pearson Education, Inc.
The function is one-to-one. Find its inverse.
a.
c. f 1 x 2x 8
4x 1f 1 x 4x 2
x 8
f 1 x 2x 2
4x 1f 1 x x 8
4x 2
f x 2x 8
4x 1
b.
d.
Slide 6 - 12Copyright © 2009 Pearson Education, Inc.
Graph
a. b.
c. d.
f x 2 x 3.
Slide 6 - 13Copyright © 2009 Pearson Education, Inc.
Graph
a. b.
c. d.
f x 2 x 3.
Slide 6 - 14Copyright © 2009 Pearson Education, Inc.
Solve the equation.
a.
b.
c.
d. 3
13
3
13
13
3
13
3
84 x 1 325 x
Slide 6 - 15Copyright © 2009 Pearson Education, Inc.
Solve the equation.
a.
b.
c.
d. 3
13
3
13
13
3
13
3
84 x 1 325 x
Slide 6 - 16Copyright © 2009 Pearson Education, Inc.
Change to an equivalent expression involving logarithms.
a.
b.
c.
d. log5 x 3
log3 x 5
logx 5 3
log5 3 x
35 x
Slide 6 - 17Copyright © 2009 Pearson Education, Inc.
Change to an equivalent expression involving logarithms.
a.
b.
c.
d. log5 x 3
35 x
log3 x 5
logx 5 3
log5 3 x
Slide 6 - 18Copyright © 2009 Pearson Education, Inc.
Find the exact value of
a.
b.
c.
d. 3
log4
1
64.
1
3
3
1
3
Slide 6 - 19Copyright © 2009 Pearson Education, Inc.
Find the exact value of
a.
b.
c.
d. 3
log4
1
64.
1
3
3
1
3
Slide 6 - 20Copyright © 2009 Pearson Education, Inc.
Solve the equation.
a.
b.
c.
d. 8, 9
log72 x2 x 1
8,9
8,9
1, 72
Slide 6 - 21Copyright © 2009 Pearson Education, Inc.
Solve the equation.
a.
b.
c.
d. 8, 9
log72 x2 x 1
8,9
8,9
1, 72
Slide 6 - 22Copyright © 2009 Pearson Education, Inc.
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.
b.
c.
d. lna lnb
a b
ab
a b
Slide 6 - 23Copyright © 2009 Pearson Education, Inc.
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.
b.
c.
d. lna lnb
a b
ab
a b
Slide 6 - 24Copyright © 2009 Pearson Education, Inc.
Write as the sum and/or difference of
a.
b.
c.
d. log19 5 log19 q log19 p
1
3log19 5 2 log19 q 2 log19 p
1
3log19 5 2 log19 q log19 p
3log19 5 2 log19 q log19 3
log19
53
q2 plogarithms. Express powers as factors.
Slide 6 - 25Copyright © 2009 Pearson Education, Inc.
Write as the sum and/or difference of
a.
b.
c.
d. log19 5 log19 q log19 p
1
3log19 5 2 log19 q 2 log19 p
1
3log19 5 2 log19 q log19 p
3log19 5 2 log19 q log19 3
log19
53
q2 plogarithms. Express powers as factors.
Slide 6 - 26Copyright © 2009 Pearson Education, Inc.
Use the Change-of-Base Formula and a calculator to evaluate
a.
b.
c.
d. 1.928
0.519
1.629
6.086
log7 42.60.
Slide 6 - 27Copyright © 2009 Pearson Education, Inc.
Use the Change-of-Base Formula and a calculator to evaluate
a.
b.
c.
d. 1.928
0.519
1.629
6.086
log7 42.60.
Slide 6 - 28Copyright © 2009 Pearson Education, Inc.
Solve the equation
a.
b.
c.
d. 27
53
3,27
log3 x log3 x 24 4.
Slide 6 - 29Copyright © 2009 Pearson Education, Inc.
Solve the equation
a.
b.
c.
d. 27
53
3,27
log3 x log3 x 24 4.
Slide 6 - 30Copyright © 2009 Pearson Education, Inc.
Solve the equation
a.
b.
c.
d. ln 6
ln 2
ln 6
ln 3
ln 2
ln 2
ln 3
32 x 3x 6 0.
Slide 6 - 31Copyright © 2009 Pearson Education, Inc.
Solve the equation
a.
b.
c.
d. ln 6
ln 2
ln 6
ln 3
ln 2
ln 2
ln 3
32 x 3x 6 0.
Slide 6 - 32Copyright © 2009 Pearson Education, Inc.
Use a graphing utility to solve the equation
a.
b.
c.
d. 0.57
1.27
2.17
1.14
ex ln x 3.
Slide 6 - 33Copyright © 2009 Pearson Education, Inc.
Use a graphing utility to solve the equation
a.
b.
c.
d. 0.57
1.27
2.17
1.14
ex ln x 3.
Slide 6 - 34Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 35Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 36Copyright © 2009 Pearson Education, Inc.
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%
Slide 6 - 37Copyright © 2009 Pearson Education, Inc.
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%
Slide 6 - 38Copyright © 2009 Pearson Education, Inc.
What principal invested 8% compounded continuously for 4 years will yield $1190?
a. $627.48
b. $1638.78
c. $1188.62
d. $864.12
Slide 6 - 39Copyright © 2009 Pearson Education, Inc.
What principal invested 8% compounded continuously for 4 years will yield $1190?
a. $627.48
b. $1638.78
c. $1188.62
d. $864.12
Slide 6 - 40Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 41Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 42Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 43Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 44Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 45Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 46Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 47Copyright © 2009 Pearson Education, Inc.
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
Slide 6 - 48Copyright © 2009 Pearson Education, Inc.
The logistic growth function
a. 600 butterflies
b. 343 butterflies
c. 360 butterflies
d. 7200 butterflies
describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. How many butterflies are expected in the habitat after 20 months?
f t 360
111e 0.27t
Slide 6 - 49Copyright © 2009 Pearson Education, Inc.
The logistic growth function
a. 600 butterflies
b. 343 butterflies
c. 360 butterflies
d. 7200 butterflies
describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. How many butterflies are expected in the habitat after 20 months?
f t 360
111e 0.27t
Slide 6 - 50Copyright © 2009 Pearson Education, Inc.
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.
a.
c.
y314.79
1 7.86e 0.246 x
y311.63
1 8.1e 0.253x
b.
d.
y314.79
1 7.86e 1.22 x
y306.53
1 7.92e 0.254 x
Time, min 5 10 15 20 25
Temperature, ºF 100 180 270 300 305
Slide 6 - 51Copyright © 2009 Pearson Education, Inc.
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.
a.
c.
y314.79
1 7.86e 0.246 x
y311.63
1 8.1e 0.253x
b.
d.
y314.79
1 7.86e 1.22 x
y306.53
1 7.92e 0.254 x
Time, min 5 10 15 20 25
Temperature, ºF 100 180 270 300 305