Deck 14: Exponential and Logarithmic Functions

Question 1

Solve the equation. $\left( 3 ^ { x + 5 } \right) \left( 3 ^ { x } \right) = 81$

A) $x = \frac { 3 } { 2 }$
B) $x = \frac { 9 } { 2 }$
C) $x = - \frac { 1 } { 2 }$
D) $x = - \frac { 7 } { 2 }$

Answer 1

$x = - \frac { 1 } { 2 }$

Question 2

An equation of the form $f ( x ) = b ^ { x }$ is a one-to-one function.

Answer 2

True

Question 3

Solve the equation. $10 ^ { x } = 0.1$

A) x = 3
B) x = $-$ 3
C) x = 1
D) x = 2
E) x = $-$ 1

Answer 3

x = $-$ 1

Question 4

Solve the equation. $10 ^ { x } = 0.000001$

A) x = 8
B) x = $-$ 6
C) x = 6
D) x = 7
E) x = $-$ 8

Question 5

The graphs of $f ( x ) = 4 ^ { x }$ and $f ( x ) = \left( \frac { 1 } { 4 } \right) ^ { x }$ are reflections of each other across the x axis.

Question 6

If $b > 0$ and $b \neq 1$ , then the function f defined by $f ( x ) = b ^ { x }$ where x is any real number, is called the power function with base b .

Question 7

Solve the equation. $\left( \frac { 1 } { 6 } \right) ^ { x } = \frac { 1 } { 36 }$

A) x = $-$ 2
B) x = 2
C) x = $-$ 4
D) x = 4
E) x = $-$ 3

Question 8

All graphs of $f ( x ) = b ^ { - x }$ contain the point $( 1,0 )$ .

Question 9

Solve the equation. $81 ^ { x } = 243$

A) $x = - \frac { 5 } { 4 }$
B) $x = \frac { 9 } { 4 }$
C) $x = \frac { 5 } { 4 }$
D) $x = \frac { 1 } { 4 }$

Question 10

Solve the equation. $\left( 256 ^ { x } \right) \left( 4 ^ { 12 x - 1 } \right) = 64$

A) $x = \frac { 1 } { 4 }$
B) $x = \frac { 9 } { 4 }$
C) $x = \frac { 5 } { 4 }$
D) $x = - \frac { 5 } { 4 }$

Question 11

Solve the equation. $\left( 3 ^ { 4 x - 4 } \right) \left( 3 ^ { x + 1 } \right) = 27$

A) $x = - \frac { 6 } { 5 }$
B) $x = \frac { 11 } { 5 }$
C) $x = - \frac { 1 } { 5 }$
D) $x = \frac { 6 } { 5 }$

Question 12

Graph the exponential function. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$

A)
B)
C)

Question 13

Solve the equation. $4 ^ { 2 x - 4 } = \frac { 1 } { 256 }$

A) x = 2
B) x = 0
C) x = 1
D) x = $-$ 2

Question 14

Solve the equation. $4 ^ { 2 x } = 16$

A) x = 1
B) x = $-$ 1
C) x = $-$ 3
D) x = 3
E) x = $-$ 2

Question 15

Solve the equation. $3 ^ { x } = 27$

A) x = $-$ 3
B) x = 4
C) x = 3
D) x = $-$ 5
E) x = 5

Question 16

Graph the exponential function. $f ( x ) = 4 ^ { x }$

A)
B)
C)

Question 17

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

A) x = 2
B) x = 4
C) x = $-$ 1
D) x = $-$ 3
E) x = 1

Question 18

Solve the equation. $27 ^ { 38 x } = 9 ^ { x + 7 }$

A) $x = - \frac { 9 } { 8 }$
B) $x = \frac { 1 } { 8 }$
C) $x = \frac { 17 } { 8 }$
D) $x = \frac { 9 } { 8 }$

Question 19

Solve the equation. $( 256 ) \left( 4 ^ { x } \right) = 64 ^ { x }$

A) x = $-$ 2
B) x = 4
C) x = $-$ 4
D) x = 3
E) x = 2

Question 20

Solve the equation. $4 ^ { 2 x - 3 } = 64$

A) x = 3
B) x = $-$ 3
C) x = $-$ 5
D) x = 5
E) x = 4

Question 21

Solve the equation.

Question 22

Solve the equation.

Question 23

Solve the equation.

Question 24

Solve the equation. Please enter your answer as a fraction.

Question 25

Use a graphing calculator to graph the exponential functions $f ( x ) = ( 2.5 ) ^ { x } \text { and } f ( x ) = ( 0.3 ) ^ { x }$ on the same set of axes. Choose the answer from the following (the blue curve represents f ( x ) = (2.5) x and the red curve represents f ( x ) = (0.3) x ).

A)
B)
C)

Question 26

Graph the exponential function. $f ( x ) = \left( \frac { 8 } { 7 } \right) ^ { x }$

A)
B)
C)

Question 27

Interest compounded annually is found using the formula $A = P ( 1 + r t )$

Question 28

Solve the equation.

Question 29

Solve the equation.

Question 30

Solve the equation.

Question 31

Graph the exponential function. $f ( x ) = 5 ^ { x + 2 }$

A)
B)
C)

Question 32

A problem compounded quarterly is found by the formula $A = p \left( 1 + \frac { r } { 4 } \right) ^ { 4 t }$ .

Question 33

Solve the equation.

Question 34

Solve the equation.

Question 35

Graph the exponential function. $f ( x ) = 5 ^ { - x - 2 }$

A)
B)
C)

Question 36

Solve the equation.

Question 37

Solve the equation. Please enter your answer as a fraction.

Question 38

Solve the equation.

Question 39

Solve the equation.

Question 40

Solve the equation.

Question 41

Graph the exponential function. $f ( x ) = - e ^ { x }$

A)
B)
C)
D)
E)

Question 42

The half-life of a radioactive substance is the amount of time that it takes for one-half of an initial amount of the substance to disappear as the result of decay.

Question 43

In a certain balloon mortgage loan, the borrower pays the lender all of the principal and interest for the loan at the end of four years. What will be the payoff amount for a loan of $185,000 at 4% annual interest rate where the interest is compounded monthly?

A) $217,041.85
B) $217,041.70
C) $217,041.80
D) $217,041.65
E) $217,041.75

Question 44

If $k > 0$ , then the formula $Q ( t ) = Q _ { 0 } e ^ { k t }$ is the law of decay model.

Question 45

Suppose it is estimated that the value of a car depreciates 30% per year for the first 10 years. The equation $A = P _ { 0 } ( 0.7 ) ^ { t }$ yields the value ( A ) of a car after t years if the original price is P o . Find the value (to the nearest dollar) of $16,000.00 car after 2 years.

A) $5,227
B) $7,840
C) $18,032
D) $3,920

Question 46

Use the formula $A = p e ^ { r t }$ to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $13,000 for 12 years at 9%. The choices are rounded to the nearest cent.

A) $38,280.83
B) $14,178.09
C) $103,358.25
D) $95,702.09
E) $15,312.33

Question 47

Use the formula $A = p \left( 1 + \frac { r } { n } \right) ^ { n t }$ to find the total amount of money accumulated at the end of the indicated time period for the following investments: $1,500 for 5 years at 4.5% compounded quarterly. The choices are rounded to the nearest cent.

A) $507.06
B) $6,941.67
C) $1,876.13
D) $750.45
E) $4,690.31

Question 48

Assuming that the rate of inflation is 4% per year, the equation P = P _o (1.04) ^t yields the predicted price P , in t years, of an item that presently costs P _o. Find the predicted price of a $0.95 soft drink in 9 years. The choices are rounded to the nearest cent.

A) $0.68
B) $1.35
C) $1.78
D) $2.20

Question 49

Mark became overextended in his gambling debt and could not pay $750 he owed. The loan person said he could have five weeks to pay off the $750 at 8% interest per week compounded continuously. How much will Mark have to pay at the end of the five weeks? Please round the answer to the nearest hundredth.

A) $1,118.87
B) $1,119.07
C) $1,119.02
D) $1,118.97
E) $1,118.92

Question 50

Use the formula $A = p e ^ { r t }$ to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $6,750 for 11 years at 9.5%. The choices are rounded to the nearest cent.

A) $7,677.18
B) $19,192.94
C) $5,187.28
D) $71,013.88
E) $47,982.35

Question 51

Use the formulas $A = p \left( 1 + \frac { r } { n } \right) ^ { n t }$ or $A = p \mathrm { e } ^ { r t }$ to find the amount for the investment. $10,800 for 5 years at 5% compounded continuously.

A) $13,867.40
B) $13,867.47
C) $13,867.54
D) $13,867.26
E) $13,867.33

Question 52

Rueben has a finance plan with the furniture store where the $4,770 he spent accrues finance charges at an annual interest rate of 9.3% compounded monthly for 4 years before he starts to make payments. What will be the balance on the account when he begins making payments?

A) $6,909.63
B) $6,909.72
C) $6,909.60
D) $6,909.69
E) $6,909.66

Question 53

Graph the exponential function. $f ( x ) = e ^ { x } + 5$

A)
B)
C)
D)
E)

Question 54

Use the formula $A = p \left( 1 + \frac { r } { n } \right) ^ { n t }$ to find the amount for the investment. $2,600 for 6 years at 3% compounded quarterly.

A) $3,110.68
B) $3,110.63
C) $3,110.53
D) $3,110.48
E) $3,110.58

Question 55

Graph the exponential function. $f ( x ) = e ^ { - x }$

A)
B)
C)
D)
E)

Question 56

Suppose that in a certain culture, the equation Q ( t ) = 1,000 e ^{0.5 t} expresses the number of bacteria present as a function of the time t , where t is expressed in hours. How many bacteria are present at the end of 2 hours? The choices are rounded to the nearest whole.

A) 2,718 bacteria
B) 1,812 bacteria
C) 1,599 bacteria
D) 4,077 bacteria
E) 4,621 bacteria

Question 57

Use the formula $A = p \left( 1 + \frac { r } { n } \right) ^ { n t }$ to find the amount for the investment. $13,400 for 4 years at 1.5% compounded semiannually.

A) $14,225.48
B) $14,225.36
C) $14,225.60
D) $14,225.54
E) $14,225.42

Question 58

Use the formula $A = p e ^ { r t }$ to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $1,000 for 2 years at 2%. The choices are rounded to the nearest cent.

A) $693.87
B) $1,561.22
C) $2,810.19
D) $1,040.81
E) $385.49

Question 59

Use the formulas $A = p \left( 1 + \frac { r } { n } \right) ^ { n t }$ or $A = p \mathrm { e } ^ { r t }$ to find the amount for the investment. $4,200 for 5 years at 5% compounded continuously.

A) $5,392.77
B) $5,392.84
C) $5,392.70
D) $5,392.63
E) $5,392.91

Question 60

Use the formula $A = p e ^ { r t }$ to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $4,600 for 6 years at 3%. The choices are rounded to the nearest cent.

A) $9,362.24
B) $3,671.47
C) $8,260.80
D) $5,507.20
E) $3,239.53

Question 61

True or false? The function below is one-to-one. f ( x ) = | x | + 1

Question 62

True or false? The graph below represents a one-to-one function.

Question 63

A function that is decreasing over its entire domain is a one-to-one function.

Question 64

Use the formulas or to find the amount for the investments. $10,500 for 5 years at 4% compounded continuously. Please round the answer to the nearest hundredth. $__________ $10,500 for 5 years at 4.5% compounded quarterly. Please round the answer to the nearest hundredth. $__________ Determine which investment amounts to more. __________

Question 65

Use the formulas or to find the amount for the investments. $4,800 for 5 years at 4% compounded continuously. Please round the answer to the nearest cent. $__________ $4,800 for 5 years at 4% compounded monthly. Please round the answer to the nearest cent. $__________ Determine which investment amounts to more. __________

Question 66

Use the formula to find the amount for the investments. $2,200 for 5 years at 1.5% compounded annually. Please round the answer to the nearest hundredth. $__________ $2,200 for 5 years at 1% compounded quarterly. Please round the answer to the nearest hundredth. $__________ Determine which investment amounts to more. __________

Question 67

Use the formula to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $2,250 for 4 years at 3%. Please round the answer to the nearest cent. $__________

Question 68

Use the formula to find the amount for the investments. $13,600 for 5 years at 4.5% compounded semiannually. Please round the answer to the nearest cent. $__________ $13,600 for 5 years at 4% compounded monthly. Please round the answer to the nearest cent. $__________ Determine which investment amounts to more. __________

Question 69

True or false? The graph below represents a one-to-one function.

Question 70

Use the formula to find the total amount of money accumulated at the end of the indicated time period for the following investment: $3,000 for 3 years at 2.5% compounded quarterly. Please round the answer to the nearest cent. $__________

Question 71

True or false? The function below is one-to-one. f ( x ) = 5 x + 3

Question 72

Use the formula to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $2,000 for 3 years at 4%. Please round the answer to the nearest cent. $__________

Question 73

True or false? The function below is one-to-one. f ( x ) = x ⁷

Question 74

True or false? The function below is one-to-one function. f ( x ) = $-$ x 8

Question 75

Use the formula to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $7,250 for 12 years at 6.5%. Please round the answer to the nearest cent. $__________

Question 76

A function that has exactly one value of x associated with each value of $f ( x )$ is a one-to-one function.

Question 77

Mark became overextended in his gambling debt and could not pay $400 he owed. The loan person said he could have three weeks to pay off the $400 at 6% interest per week compounded continuously. How much will Mark have to pay at the end of the three weeks? Please round the answer to the nearest hundredth. $__________

Question 78

Use the formula to find the total amount of money accumulated at the end of the indicated time period by compounding continuously: $13,000 for 8 years at 6.25%. Please round the answer to the nearest cent. $__________

Question 79

Rueben has a finance plan with the furniture store where the $4,990 he spent accrues finance charges at an annual interest rate of 10.3% compounded monthly for 4 years before he starts to make payments. What will be the balance on the account at the he must begin paying off the balance? Please round the answer to the nearest hundredth. $__________

Question 80

In a certain balloon mortgage loan, the borrower pays the lender all of the principal and interest for the loan at the end of three years. What will be the payoff amount for a loan of $195,000 at 4% annual interest rate where the interest is compounded monthly? Please round the answer to the nearest hundredth. $__________