Precalculus with Limits (2010)

Ron Larson, David C. Falvo

Chapter 3

Exponential and Logarithmic Functions

Educators

LW
KK
+ 2 more educators

Problem 1

Polynomial and rational functions are examples of _____________ functions.

LW
Luke W.
Numerade Educator

Problem 2

Exponential and logarithmic functions are examples of nonalgebraic functions, also called ____________ functions.

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Problem 3

You can use the ______________ Property to solve simple exponential equations.

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Problem 4

The exponential function given by $f(x)=e^{x}$ is called the _______ __________ function, and the base $e$ is called the __________ base.

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Problem 5

To find the amount $A$ in an account after $t$ years with principal $P$ and an annual interest rate $r$ compounded $n$ times per year, you can use the formula ____________

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Problem 6

To find the amount $A$ in an account after $t$ years with principal $P$ and an annual interest rate $r$ compounded continuously, you can use the formula ___________

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Problem 7

In Exercises $7-12,$ evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ {f(x)=0.9^{x}} & {x=1.4}\end{array}$$

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Problem 8

In Exercises $7-12,$ evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=2.3^{x} &x=\frac{3}{2}\end{array}$$

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Problem 9

In Exercises $7-12,$ evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=5^{x} & x=-\pi\end{array}$$

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Problem 10

In Exercises $7-12,$ evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=\left(\frac{2}{3}\right)^{5 x} & x=\frac{3}{10}\end{array}$$

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Problem 11

In Exercises $7-12,$ evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ g(x)=5000\left(2^{x}\right) & x=-1.5\end{array}$$

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Problem 12

In Exercises $7-12,$ evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=200(1.2)^{12 x} & x=24\end{array}$$

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Problem 13

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

$$f(x)=2^{x}$$

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Problem 14

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

$$f(x)=2^{x}+1$$

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Problem 15

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

$$f(x)=2^{-x}$$

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Problem 16

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

$$f(x)=2^{x-2}$$

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Problem 17

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=\left(\frac{1}{2}\right)^{x}$$

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Problem 18

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=\left(\frac{1}{2}\right)^{-x}$$

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Problem 19

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=6^{-x}$$

KK
Kalpana K.
Numerade Educator

Problem 20

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=6^{x}$$

Raushan K.
Numerade Educator

Problem 21

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=2^{x-1}$$

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Problem 22

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=4^{x-3}+3$$

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Problem 23

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

$$f(x)=3^{x}, g(x)=3^{x}+1$$

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Problem 24

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

$$f(x)=4^{x}, g(x)=4^{x-3}$$

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Problem 25

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

$$f(x)=2^{x}, g(x)=3-2^{x}$$

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Problem 26

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

$$f(x)=10^{x}, g(x)=10^{-x+3}$$

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Problem 27

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

$$f(x)=\left(\frac{7}{2}\right)^{x}, g(x)=-\left(\frac{7}{2}\right)^{-x}$$

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Problem 28

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

$$f(x)=0.3^{x}, g(x)=-0.3^{x}+5$$

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Problem 29

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

$$y=2^{-x^{2}}$$

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Problem 30

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

$$y=3^{-|x|}$$

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Problem 31

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

$$y=3^{x-2}+1$$

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Problem 32

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

$$y=4^{x+1}-2$$

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Problem 33

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ h(x)=e^{-x} & x=\frac{3}{4}\end{array}$$

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Problem 34

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=e^{x} & x=3.2\end{array}$$

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Problem 35

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=2 e^{-5 x} & x=10\end{array}$$

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Problem 36

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=1.5 e^{x / 2} & x=240\end{array}$$

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Problem 37

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=5000 e^{0.06 x} & x=6\end{array}$$

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Problem 38

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=250 e^{0.05 x} & x=20\end{array}$$

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Problem 39

In Exercises $39-44,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=e^{x}$$

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Problem 40

In Exercises $39-44,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=e^{-x}$$

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Problem 41

In Exercises $39-44,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=3 e^{x+4}$$

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Problem 42

In Exercises $39-44,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=2 e^{-0.5 x}$$

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Problem 43

In Exercises $39-44,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=2 e^{x-2}+4$$

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Problem 44

In Exercises $39-44,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

$$f(x)=2+e^{x-5}$$

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Problem 45

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

$$y=1.08^{-5 x}$$

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Problem 46

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

$$y=1.08^{5 x}$$

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Problem 47

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

$$s(t)=2 e^{0.12 t}$$

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Problem 48

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

$$s(t)=3 e^{-0.2 t}$$

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Problem 49

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

$$g(x)=1+e^{-x}$$

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Problem 50

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

$$h(x)=e^{x-2}$$

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Problem 51

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$3^{x+1}=27$$

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Problem 52

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$2^{x-3}=16$$

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Problem 53

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$\left(\frac{1}{2}\right)^{x}=32$$

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Problem 54

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$5^{x-2}=\frac{1}{125}$$

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Problem 55

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$e^{3 x+2}=e^{3}$$

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Problem 56

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$e^{2 x-1}=e^{4}$$

Charles S.
Numerade Educator

Problem 57

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$e^{x^{2}-3}=e^{2 x}$$

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Problem 58

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

$$e^{x^{2}+6}=e^{5 x}$$

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Problem 59

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested at
rate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$P=\$ 1500, r=2 \%, t=10$$

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Problem 60

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested atrate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$P=\$ 2500, r=3.5 \%, t=10$$

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Problem 61

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested atrate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$P=\$ 2500, r=4 \%, t=20$$

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Problem 62

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested atrate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$P=\$ 1000, r=6 \%, t=40$$

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Problem 63

In Exercises $63-66,$ complete the table to determine the balance $A$ for $\$ 12,000$ invested at rate $r$ for $t$ years, compounded continuously.

$$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$r=4 \%$$

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Problem 64

In Exercises $63-66,$ complete the table to determine the balance $A$ for $\$ 12,000$ invested at rate $r$ for $t$ years, compounded continuously.

$$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$r=6 \%$$

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Problem 65

In Exercises $63-66,$ complete the table to determine the balance $A$ for $\$ 12,000$ invested at rate $r$ for $t$ years, compounded continuously.

$$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$r=6.5 \%$$

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Problem 66

In Exercises $63-66,$ complete the table to determine the balance $A$ for $\$ 12,000$ invested at rate $r$ for $t$ years, compounded continuously.

$$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$r=3.5 \%$$

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Problem 67

On the day of a child's birth, a deposit of $\$ 30,000$ is made in a trust fund that pays 5$\%$ interest, compounded continuously. Determine the balance in this account on the child's 25 th birthday.

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Problem 68

A deposit of $\$ 5000$ is made in a trust fund that pays 7.5$\%$ interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive?

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Problem 69

If the annual rate of inflation averages 4$\%$ over the next 10 years, the approximate costs $C$ of
goods or services during any year in that decade will be modeled by $C(t)=P(1.04)^{t},$ where $t$ is the time in years and $P$ is the present cost. The price of an oil change for your car is presently $\$ 23.95 .$ Estimate the price 10 years from now.

Darin B.
Numerade Educator

Problem 70

The number $V$ of computers infected by a computer virus increases according to the model $V(t)=100 e^{4.4602 t}$ , where $t$ is the time in hours. Find the number of computers infected after (a) 1 hour, (b) 1.5 hours, and (c) 2 hours.

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Problem 71

The projected populations of California for the years 2015 through 2030 can be modeled by $P=34.696 e^{0.0098 t},$ where $P$ is the population (in millions) and $t$ is the time (in years), with $t=15$ corresponding to 2015 . (Source: U.S. Census Bureau)
(a) Use a graphing utility to graph the function for the years 2015 through 2030 .
(b) Use the table feature of a graphing utility to create a table of values for the same time period as in part (a).
(c) According to the model, when will the population of California exceed 50 million?

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Problem 72

The populations $P$ (in millions) of Italy from 1990 through 2008 can be approximated by the model $P=56.8 e^{0.0015 t},$ where $t$ represents the year, with $t=0$ corresponding to $1990 .$ (Source: U.S.
Census Bureau, International Data Base)
(a) According to the model, is the population of Italy increasing or decreasing? Explain.
(b) Find the populations of Italy in 2000 and 2008 .
(c) Use the model to predict the populations of Italy in 2015 and 2020 .

Gregory S.
Numerade Educator

Problem 73

Let $Q$ represent a mass of radioactive plutonium ( $^{239} \mathrm{Pu} )$ (in grams), whose half- life is $24,100$ years. The quantity of plutonium present after $t$ years is $Q=16\left(\frac{1}{2}\right)^{t / 24,100} .$
(a) Determine the initial quantity (when $t=0$ )
(b) Determine the quantity present after $75,000$ years.
(c) Use a graphing utility to graph the function over the interval $t=0$ to $t=150,000$ .

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Problem 74

Let $Q$ represent a mass of carbon 14$(14 \mathrm{C})$ (in grams), whose half-life is 5715 years.
The quantity of carbon 14 present after $t$ years is $Q=10\left(\frac{1}{2}\right)^{t / 5715}$
(a) Determine the initial quantity (when $t=0$ )
(b) Determine the quantity present after 2000 years.
(c) Sketch the graph of this function over the interval $t=0$ to $t=10,00$ .

Deboney H.
Numerade Educator

Problem 75

After $t$ years, the value of a wheel-chair conversion van that originally cost $\$ 30,500$ depreciates so that each year it is worth $\frac{7}{8}$ of its value for the previous year.
(a) Find a model for $V(t),$ the value of the van after
(b) Determine the value of the van 4 years after it was purchased.

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Problem 76

Immediately following an injection, the concentration of a drug in the bloodstream is 300 milligrams per milliliter. After $t$ hours, the the concentration is 75$\%$ of the level of the previous hour.
(a) Find a model for $C(t),$ the concentration of the drug after $t$ hours.
(b) Determine the concentration of the drug after 8 hours.

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Problem 77

TRUE OR FALSE? In Exercises 77 and $78,$ determine whether the statement is true or false. Justify your answer.

The line $y=-2$ is an asymptote for the graph of $f(x)=10^{x}-2$

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Problem 78

TRUE OR FALSE? In Exercises 77 and $78,$ determine whether the statement is true or false. Justify your answer.

$$e=\frac{271,801}{99,990}$$

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Problem 79

THINK ABOUT IT In Exercises $79-82,$ use properties of exponents to determine which functions (if any) are the same.

$$\begin{array}{l}{f(x)=3^{x-2}} \\ {g(x)=3^{x}-9} \\ {h(x)=\frac{1}{9}\left(3^{x}\right)}\end{array}$$

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Problem 80

THINK ABOUT IT In Exercises $79-82,$ use properties of exponents to determine which functions (if any) are the same.

$$\begin{array}{l}{f(x)=4^{x}+12} \\ {g(x)=2^{2 x+6}} \\ {h(x)=64\left(4^{x}\right)}\end{array}$$

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Problem 81

In Exercises $79-82,$ use properties of exponents to determine which functions (if any) are the same.

$$\begin{array}{l}{f(x)=16\left(4^{-x}\right)} \\ {g(x)=\left(\frac{1}{4}\right)^{x-2}} \\ {h(x)=16\left(2^{-2 x}\right)}\end{array}$$

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Problem 82

In Exercises $79-82,$ use properties of exponents to determine which functions (if any) are the same.

$$\begin{array}{l}{f(x)=e^{-x}+3} \\ {g(x)=e^{3-x}} \\ {h(x)=-e^{x-3}}\end{array}$$

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Problem 83

Graph the functions given by $y=3^{x}$ and $y=4^{x}$ and use the graphs to solve each inequality.
$\begin{array}{ll}{\text { (a) } 4^{x}<3^{x}} & {\text { (b) } 4^{x}>3^{x}}\end{array}$

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Problem 84

Use a graphing utility to graph each function. Use the graph to find where the function is increasing and
decreasing, and approximate any relative maximum or minimum values.
$\begin{array}{ll}{\text { (a) } f(x)=x^{2} e^{-x}} & {\text { (b) } g(x)=x 2^{3-x}}\end{array}$

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Problem 85

Use a graphing utility to graph $y_{1}=(1+1 / x)^{x}$ and $y_{2}=e$ in the same viewing window. Using the trace feature, explain what happens to the graph of $y_{1}$ as $x$ increases.

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Problem 86

Use a graphing utility to graph
$$f(x)=\left(1+\frac{0.5}{x}\right)^{x}$ and $g(x)=e^{0.5}$$
in the same viewing window. What is the relationship between $f$ and $g$ as $x$ increases and decreases without bound?

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Problem 87

Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs.
$\begin{array}{lll}{\text { (a) } y_{1}=2^{x}, y_{2}=x^{2}} & {\text { (b) } y_{1}=3^{x}, y_{2}=x^{3}}\end{array}$

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Problem 88

THINK ABOUT IT Which functions are exponential?
$\begin{array}{llll}{\text { (a) } 3 x} & {\text { (b) } 3 x^{2}} & {\text { (c) } 3^{x}} & {\text { (d) } 2^{-x}}\end{array}$

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Problem 89

Use the formula
$$A=P\left(1+\frac{r}{n}\right)^{n t}$$
to calculate the balance of an account when $P=\$ 3000$ , $r=6 \%,$ and $t=10$ years, and compounding is done
(a) by the day, (b) by the hour, (c) by the minute, and (d) by the second. Does increasing the number of
compoundings per year result in unlimited growth of the balance of the account? Explain.

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Problem 90

The figure shows the graphs of $y=2^{x}$ , $y=e^{x}, y=10^{x}, y=2^{-x}, y=e^{-x},$ and $y=10^{-x}$ . Match each function with its graph. [The graphs are labeled (a) through (f).] Explain your reasoning.

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