Deck 5: Exponential and Logarithmic Functions and Equations

Question 1

Use transformations to graph the function. Determine the domain, range, and horizontal asymptote of the function.
$$f ( x ) = 2 ( x - 3 )$$

A) domain: $( - \infty , \infty )$ ; range: $( 0 , \infty )$
horizontal asymptote: $\mathrm { y } = 0$

B) domain: $( - \infty , \infty )$ ; range: $( 0 , \infty )$
horizontal asymptote: $\mathrm { y } = 0$

C) domain: $( - \infty , \infty )$ ; range: $( - \infty , 0 )$
horizontal asymptote: $\mathrm { y } = 0$

D) domain: $( - \infty , \infty )$ ; range: $( - \infty , 0 )$
horizontal asymptote: $\mathrm { y } = 0$

Answer 1

A

Question 2

Use transformations to graph the function.
$f(x)=5(x-3)-2$

A)


B)

C)

D)

Answer 2

A

Question 3

Use transformations to graph the function.
$f(x)=e^{x}+5$

A)

B)

C)

D)

Answer 3

C

Question 4

Approximate the value using a calculator. Round your answer to three decimal places.
$$e ^ { - 1.1 }$$

A)0.633
B)-2.990
C)-0.333
D)0.333

Question 5

Convert the equation to the standard form for a hyperbola by completing the square.
A)

B)

C)

D)

Question 6

Approximate the value using a calculator. Round your answer to three decimal places.
$$\mathrm { e } ^ { 3.3 }$$

A)25.672
B)27.113
C)8.970
D)15.154

Question 7

Convert the equation to the standard form for a hyperbola by completing the square.

A) $f ( x ) = 5 ^ { x }$
B) $f ( x ) = \left( \frac { 1 } { 25 } \right) ^ { x }$
C) $f ( x ) = 25 ^ { x }$
D) $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$

Question 8

Convert the equation to the standard form for a hyperbola by completing the square.

A) $f ( x ) = \left( \frac { 4 } { 5 } \right) ^ { x }$
B) $f ( x ) = \left( \frac { 5 } { 4 } \right) ^ { x }$
C) $f ( x ) = - 2 ^ { x }$
D) $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$

Question 9

Convert the equation to the standard form for a hyperbola by completing the square.

A) $f ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }$
B) $f ( x ) = - 3 ^ { x }$
C) $f ( x ) = 4 ^ { x }$
D) $f ( x ) = - 4 ^ { x }$

Question 10

Convert the equation to the standard form for a hyperbola by completing the square.

A) $f ( x ) = 5 ^ { x }$
B) $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$
C) $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$
D) $f ( x ) = 2 ^ { x }$

Question 11

Convert the equation to the standard form for a hyperbola by completing the square.

A) $f ( x ) = 8 ^ { x }$
B) $f ( x ) = \left( \frac { 1 } { 8 } \right) ^ { x }$
C) $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$
D) $f ( x ) = 5 ^ { x }$

Question 12

Use transformations to graph the function. Determine the domain, range, and horizontal asymptote of the function.
$$f ( x ) = - 2 ^ { x + 3 } + 4$$

A) domain: $( - \infty , \infty )$ ; range: $( - \infty , - 4 )$
horizontal asymptote: $\mathrm { y } = - 4$

B) domain: $( - \infty , \infty )$ ; range: $( - 4 , \infty )$
horizontal asymptote: $\mathrm { y } = 4$

C) domain: $( - \infty , \infty )$ ; range: $( - \infty , - 4 )$
horizontal asymptote: $y = - 4$

D) domain: $( - \infty , \infty )$ ; range: $( - \infty , 4 )$
horizontal asymptote: $\mathrm { y } = 4$

Question 13

Convert the equation to the standard form for a hyperbola by completing the square.

A) $f ( x ) = - 12 ^ { x }$
B) $f ( x ) = \left( \frac { 1 } { 64 } \right) ^ { x }$
C) $f ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }$
D) $f ( x ) = - 3 ^ { x }$

Question 14

Use transformations to graph the function. Determine the domain, range, and horizontal asymptote of the function.
$$f ( x ) = 3 ^ { - x } + 4$$

A) domain: $( - \infty , \infty )$ ; range: $( 4 , \infty )$
horizontal asymptote: $\mathrm { y } = 4$

B) domain: $( - \infty , \infty )$ ; range: $( 3 , \infty )$
horizontal asymptote: $\mathrm { y } = 3$

C) domain: $( - \infty , \infty )$ ; range: $( 3 , \infty )$
horizontal asymptote: $y = 3$

D) domain: $( - \infty , \infty )$ ; range: $( 4 , \infty )$
horizontal asymptote: $y = 4$

Question 15

Graph the function.
A)

B)


C)

D)

Question 16

Convert the equation to the standard form for a hyperbola by completing the square.

A) $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$
B) $f ( x ) = 9 ^ { x }$
C) $f ( x ) = 5 ^ { x }$
D) $f ( x ) = \left( \frac { 1 } { 9 } \right) ^ { x }$

Question 17

Use transformations to graph the function.
$f(x)=2^{-x}-4$

A)

B)

C)

D)

Question 18

Graph the function.
A)

B)

C)

D)

Question 19

Approximate the value using a calculator. Round your answer to three decimal places.
$$\mathrm { e } ^ { \pi }$$

A)23.141
B)22.459
C)8.540
D)5.860

Question 20

Use transformations to graph the function.
$f(x)=e^{x-5}$

A)

B)

C)

D)

Question 21

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$e ^ { 4 x - 1 } = \left( e ^ { 6 } \right) ^ { - x }$$

A) $\{ 0 \}$
B) $\left\{ \frac { 7 } { 5 } \right\}$
C) $\left\{ \frac { 1 } { 10 } \right\}$
D) $\left\{ - \frac { 1 } { 2 } \right\}$

Question 22

Solve the problem.
Laura borrows $5000 at a rate of 9% compounded monthly. Find how much Laura owes at the end of 4 years. Use: $A = P \left( 1 + \frac { r } { n } \right) ^ { \text {nt } }$ where:
$\mathrm { A } =$ final amount
$\mathrm { P } = \$ 5000$ (the amount borrowed)
$r = 9 \% = 0.09$ (the annual rate of interest)
$\mathrm { n } = 12$ (the number of times interest is compounded each year)
$t = 4$ (the duration of the loan in years)

A)$241,800.00
B)$7157.03
C)$7872.73
D)$5151.70

Question 23

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$\left( \frac { 1 } { 3 } \right) ^ { 6 x + 4 } = 9 ^ { x - 3 }$$

A) $\left\{ \frac { 5 } { 3 } \right\}$
B) $\left\{ \frac { 1 } { 4 } \right\}$
C) $\left\{ - \frac { 1 } { 7 } \right\}$
D) $\left\{ - \frac { 1 } { 8 } \right\}$

Question 24

Solve the problem.
The formula $\mathrm { P } = 14.7 \mathrm { e } ^ { - 0.21 \mathrm { x } }$ gives the average atmospheric pressure, $\mathrm { P }$ , in pounds per square inch, at an altitude $x$ , in miles above sea level. Find the average atmospheric pressure for an altitude of $2.3$ miles. Round your answer to the nearest tenth.

A) $9.1 \mathrm { lb } /$ in. $^ { 2 }$
B) $11.0 \mathrm { lb } / \mathrm { in } ^ { 2 }$
C) $8.4 \mathrm { lb } / \mathrm { in } ^ { 2 }$
D) $7.8 \mathrm { lb } /$ in. $^ { 2 }$

Question 25

Solve the problem.
Find the amount in a savings account at the end of 4 years if the amount originally deposited is $9000 and the interest rate is 5.5% compounded monthly. Use: $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ where:
$A =$ final amount
$\mathrm { P } = \$ 9000$ (the initial deposit)
$\mathrm { r } = 5.5 \% = 0.055$ (the annual rate of interest)
$\mathrm { n } = 12$ (the number of times interest is compounded each year)
$t = 4$ (the duration of the deposit in vears)

A)$433,980.00
B)$12,329.96
C)$9166.14
D)$11,209.06

Question 26

Solve the problem.
The function D(h)= 8e-0.4h can be used to determine the milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given. How many milligrams (to two decimals)will be present after 12 hours?

A)4.76 mg
B)0.07 mg
C)972.08 mg
D)0.49 mg

Question 27

Solve the problem.
The rabbit population in a forest area grows at the rate of 7% monthly. If there are 270 rabbits in September, find how many rabbits (rounded to the nearest whole number)should be expected by next September. Use $$\mathrm { v } = 270 ( 2.7 ) ^ { 0.07 \mathrm { t } }$$

A)609
B)612
C)622
D)635

Question 28

Solve the problem.
Four bacteria are placed in a petri dish. The population will triple every day. The formula for the number of bacteria in the dish on day t $$N ( t ) = 4 ( 3 ) ^ { t }$$ , where t is the number of days after the four bacteria are placed in the dish. How many bacteria are in the dish six days after the four bacteria are placed in the dish?

A)864
B)72
C)13
D)2916

Question 29

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$2 ^ { 1 + 2 x } = 8$$

A) $\{ 1 \}$
B) $\{ 4 \}$
C) $\{ - 1 \}$
D) $\{ 2 \}$

Question 30

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$32 ^ { x - 1 } = 64 ^ { 3 x }$$

A) $\left\{ - \frac { 5 } { 13 } \right\}$
B) $\left\{ - \frac { 5 } { 4 } \right\}$
C) $\left\{ - \frac { 1 } { 2 } \right\}$
D) $\left\{ - \frac { 1 } { 13 } \right\}$

Question 31

Solve the problem.
A city is growing at the rate of 0.9% annually. If there were 5,400,000 residents in the city in 1992, find how many (to the nearest ten-thousand)were living in that city in 2000. $$\text { Use } y = 5,400,000 ( 2.7 ) ^ { 0.009 t }$$

A)14,580,000
B)5,800,000
C)5,830,000
D)1,050,000

Question 32

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$f(x)=2-e^{-x}$

A)

B)

C)

D)

Question 33

Solve the problem.
The amount of a radioactive substance present, in grams, at time t in months is given by the formula y = 8000(2)-0.3t. Find the number of grams present in 2 years. If necessary, round to three decimal places.

A)527.803
B)5278.032
C)54.409
D)5.441

Question 34

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$3 ^ { 6 - 3 x } = \frac { 1 } { 27 }$$

A) $\left\{ \frac { 1 } { 9 } \right\}$
B) $\{ 9 \}$
C) $\{ 3 \}$
D) $\{ - 3 \}$

Question 35

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$3 ( 10 - 2 x ) = 81$$

A) $\{ - 3 \}$
B) $\{ 3 \}$
C) $\{ 2 \}$
D) $\{ 4 \}$

Question 36

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$4 ( 3 x - 5 ) = 256$$

A) $\{ 128 \}$
B) $\{ 3 \}$
C) $\{ - 3 \}$
D) $\left\{ \frac { 1 } { 64 } \right\}$

Question 37

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$5 ^ { x } = 625$$

A) $\{ 4 \}$
В) $\{ 125 \}$
C) $\{ 3 \}$
D) $\{ 5 \}$

Question 38

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$2 ^ { x } = \frac { 1 } { 8 }$$

A) $\{ 3 \}$
B) $\left\{ \frac { 1 } { 3 } \right\}$
C) $\left\{ \frac { 1 } { 4 } \right\}$
D) $\{ - 3 \}$

Question 39

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$$e ^ { x - 4 } = \left( \frac { 1 } { e ^ { 3 } } \right) ^ { x + 3 }$$

A) $\left\{ \frac { 7 } { 4 } \right\}$
B) $\left\{ - \frac { 13 } { 2 } \right\}$
C) $\left\{ - \frac { 5 } { 4 } \right\}$
D) $\left\{ - \frac { 7 } { 2 } \right\}$

Question 40

Determine the correct exponential function of the form f(x)= bx whose graph is given.
$\left( e ^ { x } \right) ^ { x } \cdot e ^ { 32 } = e ^ { 12 x }$

A) $\{ 4 \}$
B) $\{ 4,8 \}$
C) $\{ 8 \}$
D) $\{ - 4 , - 8 \}$

Question 41

Solve the equation.
$$5 ^ { 3 } = x$$

A) $\log _ { 5 } x = 3$
B) $\log _ { 3 } x = 5$
C) $\log _ { 5 } 3 = x$
D) $\log _ { x } 5 = 3$

Question 42

Solve the equation.
Which has a lower present value: $30,000 if interest is paid at a rate of 5.98% compounded continuously for 5 years, or $33,000 if interest is paid at a rate of 3.3% compounded continuously for 65 months?

A)$33,000 with interest is paid at a rate of 3.3% compounded continuously for 65 months has a lower present value.
B)$30,000 with interest paid at a rate of 5.98% compounded continuously for 5 years has a lower present value.
C)Both investments have the same present value.

Question 43

Solve the equation.
$$4 ^ { - 2 } = \frac { 1 } { 16 }$$

A) $\log _ { 4 } \frac { 1 } { 16 } = - 2$
B) $\log _ { 1 / 16 } 4 = - 2$
C) $\log _ { 4 } - 2 = \frac { 1 } { 16 }$
D) $\log _ { - 2 } \frac { 1 } { 16 } = 4$

Question 44

Solve the equation.
$$4 ^ { 5 / 2 } = 32$$

A) $\log _ { 32 } 4 = \frac { 5 } { 2 }$
B) $\log _ { 4 } 32 = \frac { 5 } { 2 }$
C) $\frac { \log _ { 2 } 32 } { \log _ { 5 } 4 } = 4$
D) $\log _ { 5 } 4 = \frac { 5 } { 2 }$

Question 45

Solve the problem.
Instruments on a satellite measure the amount of power generated by the satellite's power supply. The time t and the power P can be modeled by the funct $$P = 50 e ^ { - t / 300 }$$ , where t is in days and P is in watts. How much power will be available after 378 days? Round to the nearest hundredth.

A)14.32 watts
B)4.11 watts
C)14.18 watts
D)15.95 watts

Question 46

Write the logarithmic equation as an exponential equation.
$$\log _ { 9 } \frac { 1 } { 729 }$$

A)-3
B)81
C)3
D)-81

Question 47

Write the exponential equation as an equation involving a logarithm.
$\log _ { b } 36 = 2$

A) $2 ^ { b } = 36$
B) $36 ^ { 2 } = b$
C) $b ^ { 2 } = 36$
D) $36 ^ { b } = 2$

Question 48

Write the logarithmic equation as an exponential equation.
$\log _ { 3 } 27 = x$

A) $27 ^ { 3 } = x$
B) $x ^ { 3 } = 27$
C) $27 ^ { x } = 3$
D) $3 ^ { \mathrm { x } } = 27$

Question 49

Solve the problem.
An initial investment of $1740 is appreciated for 14 years in an account that earns 7% interest, compounded continuously. Find the amount of money in the account at the end of the period. Round to the nearest cent.

A)$6,127,559.21
B)$12,856.96
C)$4636.15
D)$2868.78

Question 50

Solve the equation.
$$7 ^ { 3 } = 343$$

A) $\log _ { 7 } 3 = 343$
B) $\log _ { 7 } 343 = 3$
C) $\log _ { 343 } 7 = 3$
D) $\log _ { 3 } 343 = 7$

Question 51

Solve the problem.
Suppose your great-great grandmother invested $700 earning 3.6% interest compounded continuously 120 years ago. How much would her investment be worth today? Round to the nearest cent.

A)$52,632.04
B)$52,620.80
C)$818.18
D)$52,620.83

Question 52

Write the logarithmic equation as an exponential equation.
$$\log _ { 4 } 16$$

A)8
B)16
C)4
D)2

Question 53

Solve the problem.
An original investment of $7000 earns 7% interest compounded continuously. What will the investment be worth in 3 years? Round to the nearest cent.

A)$8470.00
B)$8635.75
C)$8735.75
D)$8575.30

Question 54

Solve the problem.
A rumor is spread at an elementary school with 1200 students according to the model $$\mathrm { N } = 1200 \left( 1 - \mathrm { e } ^ { - 0.16 \mathrm {~d} } \right)$$ where N is the number of students who have heard the rumor and d is the number of days that have elapsed since the rumor began. How many students will have heard the rumor after 5 days?

A)1063 students
B)1006 students
C)661 students
D)689 students

Question 55

Write the exponential equation as an equation involving a logarithm.
$\log _ { 5 } 25 = 2$

A) $5 ^ { 25 } = 2$
B) $5 ^ { 2 } = 25$
C) $25 ^ { 2 } = 5$
D) $2 ^ { 5 } = 25$

Question 56

Solve the problem.
Suppose that you have $12,000 to invest. Which investment yields the greater return over 7 years: 8.75% compounded continuously or 8.9% compounded semiannually?

A)$12,000 invested at 8.9% compounded semiannually over 7 years yields the greater return.
B)$12,000 invested at 8.75% compounded continuously over 7 years yields the greater return.
C)Both investment plans yield the same return.

Question 57

Write the exponential equation as an equation involving a logarithm.
$$9 ^ { \mathrm { X } } = 81$$

A) $\log _ { X } 81 = 9$
B) $\log _ { 81 } 9 = x$
C) $\log _ { 81 } x = 9$
D) $\log _ { 9 } 81 = x$

Question 58

Write the logarithmic equation as an exponential equation.
$$\log _ { 9 } \frac { 1 } { 81 }$$

A)9
B)-2
C)2
D)-9

Question 59

Write the exponential equation as an equation involving a logarithm.
$\log _ { 2 } x = 3$

A) $x ^ { 3 } = 2$
B) $2 ^ { x } = 3$
C) $3 ^ { 2 } = x$
D) $2 ^ { 3 } = x$

Question 60

Solve the problem.
Suppose that $6000 is invested at an interest rate of 5.2% per year, compounded continuously. What is the balance after 2 years?Round to the nearest cent.

A)$6312.00
B)$6757.60
C)$6657.60
D)$6624.00

Question 61

Write the logarithmic equation as an exponential equation.
ln $\mathrm { e } ^ { \sqrt { 2 } }$

A) 2
B) $\sqrt { 2 }$
C) 4
D) e

Question 62

Write the exponential equation as an equation involving a common logarithm or a natural logarithm.
$\log B = C$

A) $e ^ { B } = C$
B) $10 \mathrm { C } = \mathrm { B }$
C) $e ^ { C } = B$
D) $10 ^ { \mathrm { B } } = \mathrm { C }$

Question 63

Write the logarithmic equation as an exponential equation.
$\log 10,000$

A) $\frac { 1 } { 10000 }$
B) 40
C) $- 4$
D) 4

Question 64

Write the exponential equation as an equation involving a common logarithm or a natural logarithm.
$$\mathrm { e } ^ { - 3 } = \mathrm { t }$$

A) $\log _ { - 3 } \mathrm { t } = \mathrm { e }$
B) $\ln t = - 3$
C) $\log _ { - 3 } \mathrm { e } = \mathrm { t }$
D) $\ln ( - 3 ) = t$

Question 65

Write the logarithmic equation as an exponential equation.
$$\ln \mathrm { e } ^ { 10 }$$

A)e
B)10
C)1 $$\text { D) } \frac { 1 } { 10 }$$

Question 66

Write the logarithmic equation as an exponential equation.
ln l

A)-1
B)1
C)0
D)e

Question 67

Write the logarithmic equation as an exponential equation.
ln e

A)0
B)1
C)-1
D)e

Question 68

Write the exponential equation as an equation involving a common logarithm or a natural logarithm.
$\ln 1 = 0$

A) $\mathrm { e } ^ { 1 } = 1$
B) $\mathrm { e } ^ { 1 } = 0$
C) $e ^ { 0 } = 1$
D) $1 ^ { \mathrm { e } } = 0$

Question 69

Write the logarithmic equation as an exponential equation.
$\mathrm { e } ^ { \ln 8 }$

A) 7
B) $e ^ { 8 }$
C) 8
D) $\ln 8$

Question 70

Write the logarithmic equation as an exponential equation.
$$\log _ { 1 / 5 } 625$$

A) $- \frac { 1 } { 4 }$
B) $\frac { 1 } { 4 }$
C) $- 4$
D) 4

Question 71

Evaluate the logarithm without the use of a calculator.
$$\log _ { 3 } 3 ^ { - 4 }$$

A)1
B)3
C)-12
D)-4

Question 72

Use the properties of logarithms to evaluate the expression without the use of a calculator.
$$4 ^ { \log _ { 4 } 6 }$$

A)1
B)6
C)4
D)24

Question 73

Use the properties of logarithms to evaluate the expression without the use of a calculator.
$$12 ^ { \mathrm { X } } = 144$$

A) $\log _ { x } 144 = 12$
B) $\log _ { 12 } 144 = x$
C) $\log x = 12$
D) $\log _ { 144 } 12 = x$

Question 74

Write the logarithmic equation as an exponential equation.
$$\log _ { 3 } 1$$

A)1 $$\text { B) } \frac { 1 } { 3 }$$
C)0
D)3

Question 75

Evaluate the logarithm without the use of a calculator.
$\log _ { 7 } \sqrt { 7 }$

A) 7
B) $\frac { 1 } { 7 }$
C) $\frac { 1 } { 2 }$
D) 1

Question 76

Write the logarithmic equation as an exponential equation.
$\log ( 100 ) = 2$

A) $10 ^ { 100 } = 2$
B) $10 ^ { 1 / 2 } = 100$
C) $10 ^ { 2 } = 100$
D) $10 ^ { 20 } = 100$

Question 77

Write the exponential equation as an equation involving a common logarithm or a natural logarithm.
$\ln x = 5$

A) $e ^ { x } = 5$
B) $e ^ { 5 } = x$
C) $x ^ { 5 } = e$
D) $5 ^ { \mathrm { e } } = \mathrm { x }$

Question 78

Use the properties of logarithms to evaluate the expression without the use of a calculator.
$$\mathrm { e } ^ { \mathrm { x } } = 23$$

A) $\log _ { 23 } x = e$
B) $\ln x = 23$
C) $\log _ { x } e = 23$
D) $\ln 23 = x$

Question 79

Write the exponential equation as an equation involving a common logarithm or a natural logarithm.
$\ln \left( \frac { 1 } { \mathrm { e } } \right) = - 1$

A) $e ^ { - 1 } = - 1$
B) $e ^ { - 1 } = - e$
C) $e ^ { - 1 } = \frac { 1 } { 0 }$
D) $\mathrm { e } ^ { 1 / \mathrm { e } } = - 1$

Question 80

Use the properties of logarithms to evaluate the expression without the use of a calculator.
$$10 ^ { 2 } = 100$$

A) $\log _ { 2 } 10 = 100$
B) $\log 2 = 100$
C) $\log _ { 2 } 100 = 10$
D) $\log 100 = 2$