Deck 9: Inequalities and Problem Solving

Question 1

The function f $$f ( x ) = 800 ( 0.5 ) ^ { x / 60 }$$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where x is the number of years since the material was put into the vault. Find the amount of
Radioactive material in the vault after 40 years. Round to the nearest whole number.

A) 267 pounds
B) 600 pounds
C) 283 pounds
D) 504 pounds

Answer 1

504 pounds

Question 2

Graph the function by making a table of coordinates.
$f(x)=\left(\frac{4}{5}\right)^{x}$

A)
B)
C)
D)

Answer 2

Question 3

The graph of an exponential function is given. Select the function for the graph from the functions listed.

A) $f ( x ) = 3 ^ { x }$
B) $f ( x ) = 3 ^ { - x }$
C) $f ( x ) = - 3 ^ { x }$
D) $f ( x ) = - 3 ^ { - x }$

Answer 3

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

Question 4

Approximate the number using a calculator. Round the answer to three decimal places.
$$4 ^ { 1.1 }$$

A) 1.464
B) 4.895
C) 4.400
D) 4.595

Question 5

Graph the function.
Use the graph of $f ( x ) = 3 ^ { x }$ to obtain the graph of $g ( x ) = 3 ^ { x } + 3 + 1$

A)
B)
C)
D)

Question 6

The formula S $$S = A \left( \frac { ( 1 + r ) ^ { t + 1 } - 1 } { r } \right)$$ models the value of a retirement account, where A = the number of dollars added to the retirement account each year, r = the annual interest rate, and S = the value of the retirement
Account after t years. If the interest rate is 11%, how much will the account be worth after 15 years if $200 is
Added each year? Round to the nearest whole number.

A) $7838
B) $1053
C) $15,311
D) $3200

Question 7

Graph the function by making a table of coordinates.
$f(x)=\left(\frac{1}{2}\right)^{x}$

A)
B)
C)
D)

Question 8

Approximate the number using a calculator. Round the answer to three decimal places.
$$5 ^ { - 1.6 }$$

A) 0.076
B) -10.486
C) -8.000
D) 0.376

Question 9

Graph the function by making a table of coordinates.
$$f ( x ) = 0.9 ^ { x }$$

A)
B)
C)
D)

Question 10

The graph of an exponential function is given. Select the function for the graph from the functions listed.

A) $f ( x ) = 5 ^ { x } - 1$
B) $f ( x ) = 5 ^ { x - 1 }$
C) $f ( x ) = 5 ^ { x }$
D) $f ( x ) = 5 ^ { x } + 1$

Question 11

Approximate the number using a calculator. Round the answer to three decimal places.
$$6^{\sqrt{3}}$$

A) 22.274
B) 108.000
C) 27.000
D) 10.392

Question 12

Graph the function.
Use the graph of $f ( x ) = 4 ^ { x }$ to obtain the graph of $g ( x ) = 4 ^ { x - 3 }$

A)A)
B)B)
C)C)
D)D)

Question 13

The graph of an exponential function is given. Select the function for the graph from the functions listed.

A) $f ( x ) = 5 ^ { x }$
B) $f ( x ) = - 5 ^ { - x }$
C) $f ( x ) = 5 ^ { - x }$
D) $f ( x ) = - 5 ^ { x }$

Question 14

Graph the function by making a table of coordinates.
$f(x)=4^{x}$

A)
B)
C)
D)

Question 15

The graph of an exponential function is given. Select the function for the graph from the functions listed.

A) $f ( x ) = - 5 ^ { - x }$
B) $f ( x ) = 5 ^ { x }$
C) $f ( x ) = 5 ^ { - x }$
D) $f ( x ) = - 5 ^ { x }$

Question 16

The graph of an exponential function is given. Select the function for the graph from the functions listed.

A) $f ( x ) = 5 ^ { x - 2 }$
B) $f ( x ) = 5 ^ { x } + 2$
C) $f ( x ) = 5 ^ { x }$
D) $f ( x ) = 5 ^ { x } - 2$

Question 17

Solve.
The rabbit population in a forest area grows at the rate of 8% monthly. If there are 260 rabbits in September, find how many rabbits (rounded to the nearest whole number) should be expected by next September. Use the
Function f(x) $$f ( x ) = 260 ( 2.7 ) ^ { 0.08 t }$$

A) 688 rabbits
B) 662 rabbits
C) 674 rabbits
D) 675 rabbits

Question 18

The graph of an exponential function is given. Select the function for the graph from the functions listed.

A) $f ( x ) = 4 ^ { x } + 2$
B) $f ( x ) = 4 ^ { x }$
C) $f ( x ) = 4 ^ { x } - 2$
D) $f ( x ) = 4 ^ { x - 2 }$

Question 19

Graph the function.
Use the graph of $f ( x ) = 5 ^ { x }$ to obtain the graph of $g ( x ) = 5 ^ { x } + 2$

A)
B)
C)
D)

Question 20

A city has been growing at a rate of 0.5% annually. If there are currently 4,457,000 residents in the city, how many (to the nearest ten-thousand) would be living in this city six years from now? Use the function $$f ( x ) = 4,457,000 ( 2.7 ) ^ { 0.005 t }$$

A) 4,620,000
B) 4,590,000
C) 360,000
D) 12,030,000

Question 21

Solve the problem.
The growth in the mouse population at a certain county dump can be modeled by the exponential function $$\mathrm { A } ( \mathrm { t } ) = 224 \mathrm { e } ^ { 0.019 \mathrm { t } }$$ where t is the number of months since the population was first recorded. Estimate the population after 28 months.

A) 389 mice
B) 381 mice
C) 191 mice
D) 228 mice

Question 22

Graph the function.
$$\text { Use the graph of } f ( x ) = 3 ^ { x } \text { to obtain the graph of } g ( x ) = 3 ^ { x } - 2$$

A)
B)
C)
D)

Question 23

Write the equation in its equivalent exponential form.
$$\log _ { 4 } 256 = x$$

A) $256 ^ { 4 } = x$
B) $256 ^ { x } = 4$
C) $4 ^ { x } = 256$
D) $x ^ { 4 } = 256$

Question 24

Use the compound interest formulas A $$A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$$ and A = Pert to solve.
Find the accumulated value of an investment of $20,000 at 5.75% compounded annually for 7 years.

A) $27,971.27
B) $29,579.62
C) $26,900.00
D) $28,050.00

Question 25

Use the compound interest formulas A $$A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$$ and A = Pert to solve.
Find the accumulated value of an investment of $5000 at 7% compounded continuously for 6 years.

A) $7709.81
B) $7609.81
C) $7100.00
D) $7503.65

Question 26

Use the compound interest formulas A $$A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$$ and A = Pert to solve.
Find the accumulated value of an investment of $1080 at 6.5% compounded annually for 7 years.

A) $1575.87
B) $1501.20
C) $1678.31
D) $1571.40

Question 27

Use the compound interest formulas A $$A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$$ and A = Pert to solve.
Suppose that you have $9000 to invest. Which investment yields the greater return over 7 years: 7.5% compounded continuously or 7.6% compounded semiannually?

A) $9000 invested at 7.5% compounded continuously over 7 years yields the greater return.
B) Both investment plans yield the same return.
C) $9000 invested at 7.6% compounded semiannually over 7 years yields the greater return.

Question 28

Write the equation in its equivalent exponential form.
$$\log _ { 4 } 64 = 3$$

A) $4 ^ { 64 } = 3$
B) $3 ^ { 4 } = 64$
C) $64 ^ { 3 } = 4$
D) $4 ^ { 3 } = 64$

Question 29

Solve the problem.
The function D(h) = 5e-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 will be present after 12 hours?
Round the answer to two decimal places.

A) 0.04 mg
B) 607.55 mg
C) 1.62 mg
D) 2.97 mg

Question 30

Graph the function.
$$\text { Use the graph of } f ( x ) = 4 ^ { x } \text { to obtain the graph of } g ( x ) = \frac { 1 } { 4 } \cdot 4 ^ { x }$$

A)
B)
C)
D)

Question 31

Graph the function.
Use the graph of $f ( x ) = 3 ^ { x }$ to obtain the graph of $g ( x ) = 3 ^ { - x }$ .

A)
B)
C)
D)

Question 32

Graph the function.
$$\text { Use the graph of } f ( x ) = 2 ^ { x } \text { to obtain the graph of } g ( x ) = 2 ^ { x } + 1 \text {. }$$

A)
B)
C)
D)

Question 33

Graph the function.
Use the graph of $f ( x ) = 5 ^ { x }$ to obtain the graph of $g ( x ) = - 5 ^ { x }$ .

A)
B)
C)
D)

Question 34

Solve the problem.
A sample of 700 g of lead-210 decays to polonium-210 according to the function given by A(t) = 700e-0.032t, where t is time in years. What is the amount of the sample after 10 years? Round the answer to the nearest gram.

A) 964 g
B) 508 g
C) 369 g
D) 11 g

Question 35

Write the equation in its equivalent exponential form.
$$3 = \log _ { b ^ { 2 } } 216$$

A) $216 ^ { 3 } = b$
B) $3 ^ { b } = 216$
C) $b ^ { 3 } = 216$
D) $216 ^ { b } = 3$

Question 36

Approximate the number using a calculator. Round the answer to three decimal places.
$\mathrm { e } ^ { 1.3 }$

A) $3.669$
B) $3.969$
C) $3.534$
D) $2.040$

Question 37

Use the compound interest formulas A $$A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$$ and A = Pert to solve.
Find the accumulated value of an investment of $1800 at 8% compounded quarterly for 2 years.

A) $1872.72
B) $2099.52
C) $2088.00
D) $2108.99

Question 38

Use the compound interest formulas A $$A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$$ and A = Pert to solve.
Find the accumulated value of an investment of $17,000 at 6.5% compounded semiannually for 6 years.

A) $23,630.00
B) $24,805.42
C) $24,953.40
D) $20,596.30

Question 39

Write the equation in its equivalent exponential form.
$$3 = \log _ { 2 } x$$

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

Question 40

Approximate the number using a calculator. Round the answer to three decimal places.
$$\mathrm { e } ^ { - 0.5 }$$

A) -0.607
B) 0.607
C) -1.359
D) 0.907

Question 41

Graph the function.
$f(x)=\log _{1 / 6} x$

A)
B)
C)
D)

Question 42

Write the equation in its equivalent logarithmic form.
$$2 - 2 = \frac { 1 } { 4 }$$

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

Question 43

Write the equation in its equivalent logarithmic form.
$$2 ^ { 3 } = x$$

A) $\log _ { 3 } x = 2$
B) $\log _ { 2 } x = 3$
C) $\log _ { 2 } 3 = x$
D) $\log _ { x } 2 = 3$

Question 44

Write the equation in its equivalent logarithmic form.
$$\sqrt [ 3 ] { 125 } = 5$$

A) $\log _ { 5 } 125 = 3$
B) $\log _ { 125 } 5 = \frac { 1 } { 3 }$
C) $\log _ { 125 } 3 = \frac { 1 } { 5 }$
D) $\log _ { 5 } 125 = \frac { 1 } { 3 }$

Question 45

Write the equation in its equivalent logarithmic form.
$$4 ^ { x } = 64$$

A) $\log _ { 64 } x = 4$
B) $\log _ { x } 64 = 4$
C) $\log _ { 4 } 64 = x$
D) $\log _ { 64 } 4 = x$

Question 46

Evaluate the expression without using a calculator.
$$\log _ { 3 } \frac { 1 } { 27 }$$

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

Question 47

Write the equation in its equivalent logarithmic form.
$$5 ^ { 2 } = 25$$

A) $\log _ { 25 } 5 = 2$
B) $\log _ { 5 } 25 = 2$
C) $\log _ { 2 } 25 = 5$
D) $\log _ { 5 } 2 = 25$

Question 48

Evaluate the expression without using a calculator.
$\log _ { 12 } 1$

A) 12
B) 1
C) 0
D) $\frac { 1 } { 12 }$

Question 49

Evaluate the expression without using a calculator.
$$5 ^ { \log _ { 5 } 10 }$$

A) 10
B) 5
C) $\log _ { 5 } 10$
D) 15

Question 50

Evaluate the expression without using a calculator.
$\log _ { 5 } 25$

A) 10
B) 1
C) $\frac { 1 } { 2 }$
D) 2

Question 51

Evaluate the expression without using a calculator.
$\log _ { 8 } 8$

A) 1
B) $\frac { 1 } { 8 }$
C) 0
D) 8

Question 52

Write the equation in its equivalent logarithmic form.
$4 ^ { 2 } = y$

A) $\log _ { y } 4 = 2$
B) $\log _ { 4 } y = 2$
C) $\log _ { y } 2 = 4$
D) $\log _ { 2 } y = 4$

Question 53

Evaluate the expression without using a calculator.
$\log _ { 10 } \sqrt { 10 }$

A) $\frac { 1 } { 2 }$
B) 1
C) $\frac { 1 } { 10 }$
D) 10

Question 54

Graph the function.
$f(x)=\log _{2} x$

A)
B)
C)
D)

Question 55

Evaluate the expression without using a calculator.
$$\log _ { 10 } 10,000$$

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

Question 56

Evaluate the expression without using a calculator.
$\log _ { 7 } 7 ^ { 10 }$

A) 17
B) 10
C) $\log _ { 7 } 10$
D) 7

Question 57

Write the equation in its equivalent logarithmic form.
$$8 ^ { x } = 359$$

A) $\log _ { 8 } 359 = x$
B) $\log _ { 8 } x = 359$
C) $\log _ { 359 ^ { 8 } } = x$
D) $\log _ { 359 } x = 8$

Question 58

Find the domain of the logarithmic function. Write your answer in interval notation.
$$f ( x ) = \log _ { 3 } ( x + 2 )$$

A) $( 2 , \infty )$
B) $( 3 , \infty )$
C) $( - 2 , \infty )$
D) $( - \infty , 0 ) \cup ( 0 , \infty )$

Question 59

Evaluate the expression without using a calculator.
$\log _ { 25 } 5$

A) 1
B) $\frac { 1 } { 2 }$
C) 10
D) 2

Question 60

Find the domain of the logarithmic function. Write your answer in interval notation.
$$f ( x ) = \log ( x - 6 )$$

A) $( - \infty , 0 ) \cup ( 0 , \infty )$
B) $( - 6 , \infty )$
C) $( - \infty , 6 ) \cup ( 6 , \infty )$
D) $( 6 , \infty )$

Question 61

Find the domain of the logarithmic function. Write your answer in interval notation.
$$f ( x ) = \log _ { 4 } ( x - 8 ) ^ { 2 }$$

A) $( 8 , \infty )$
B) $( - 8 , \infty )$
C) $( - \infty , 0 ) \cup ( 0 , \infty )$
D) $( - \infty , 8 ) \cup ( 8 , \infty )$

Question 62

Evaluate the expression without using a calculator.
The long jump record, in feet, at a particular school can be modeled by f(x) = 18.8 + 2.2 ln(x + 1) where x is the number of years since the school was opened. What is the record for the long jump 25 years after the school
Opened? Round the answer to the nearest tenth.

A) 21.0 ft
B) 26.0 ft
C) 25.9 ft
D) 25.8 ft

Question 63

Evaluate the expression without using a calculator.
$\log 1000$

A) $\frac { 1 } { 3 }$
B) 30
C) $\frac { 3 } { 10 }$
D) 3

Question 64

Find the domain of the logarithmic function. Write your answer in interval notation.
$$f ( x ) = \log _ { 6 } ( x + 7 ) ^ { 2 }$$

A) $( - 7 , \infty )$
B) $( - \infty , 0 ) \cup ( 0 , \infty )$
C) $( 7 , \infty )$
D) $( - \infty , - 7 ) \cup ( - 7 , \infty )$

Question 65

Find the domain of the logarithmic function. Write your answer in interval notation.
$$f ( x ) = \ln ( 4 - x )$$

A) $( - 4 , \infty )$
B) $( - \infty , 4 )$
C) $( - \infty , 0 )$
D) $( - \infty , 4 ) \cup ( 4 , \infty )$

Question 66

Evaluate the expression without using a calculator.
The pH of a solution ranges from 0 to 14. An acid has a pH less than 7. Pure water is neutral and has a pH of 7. The pH of a solution is given by pH = -log x where x represents the concentration of the hydrogen ions in the
Solution in moles per liter. Find the pH if the hydrogen ion concentration is 5 × 10-12.

A) 11.7
B) 11.3
C) 12.7
D) 12.3

Question 67

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
$\log _ { 6 } \left( \frac { 6 } { x } \right)$

A) $1 - \log _ { 6 } x$
B) $\frac { 1 } { x }$
C) $- \log _ { 6 } x$
D) 6

Question 68

Evaluate the expression without using a calculator.
$\log 10 ^ { 3 }$

A) $10 ^ { 3 }$
B) $\log 3$
C) 10
D) 3

Question 69

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
$\log _ { 2 } ( 7 \cdot 3 )$

A) $\left( \log _ { 2 } 7 \right) \left( \log _ { 2 } 3 \right)$
B) $\log _ { 2 } 7 - \log _ { 2 } 3$
C) $\log _ { 2 } 7 + \log _ { 2 } 3$
D) $\log _ { 2 } 21$

Question 70

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
$\log _ { 4 } ( 16 x )$

A) $2 x$
B) $8 + \log _ { 4 } x$
C) $2 \log _ { 4 } x$
D) $2 + \log _ { 4 } x$

Question 71

Evaluate the expression without using a calculator.
$\ln \mathrm { e } ^ { 9 }$

A) $- 9$
B) 9
C) $\frac { 1 } { 9 }$
D) $\log 9$

Question 72

Evaluate the expression without using a calculator.
$\log \left( \frac { 1 } { 10,000 } \right)$

A) $- \frac { 1 } { 4 }$
B) $\frac { 1 } { 10,000 }$
C) $- 4$
D) 4

Question 73

Evaluate the expression without using a calculator.
The pH of a solution ranges from 0 to 14. An acid has a pH less than 7. Pure water is neutral and has a pH of 7. The pH of a solution is given by pH = -log x where x represents the concentration of the hydrogen ions in the
Solution in moles per liter. Find the pH if the hydrogen ion concentration is 1 × 10-6.

A) 8
B) -8
C) 6
D) -6

Question 74

Evaluate the expression without using a calculator.
$\ln \sqrt [ 8 ] { \mathrm { e } }$

A) $8 \mathrm { e }$
B) $\frac { \mathrm { e } } { 8 }$
C) $\frac { 1 } { 8 }$
D) 8

Question 75

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
$\log _ { 8 } ( 8 x )$

A) $1 + \log _ { 8 } x$
B) $x$
C) 1
D) 8

Question 76

Evaluate the expression without using a calculator.
The formula y = 1 + 1.5 ln(x + 1) models the average number of free-throws a basketball player can make consecutively during practice as a function of time, where x is the number of consecutive days the basketball
Player has practiced for two hours. After 13 days of practice, what is the average number of consecutive free
Throws the basketball player makes?

A) 5 consecutive free throws
B) 6 consecutive free throws
C) 9 consecutive free throws
D) 8 consecutive free throws

Question 77

Evaluate the expression without using a calculator.
$\ln \mathrm { e } ^ { 4.3 }$

A) $\log 4.3$
B) $4.3$
C) $- 4.3$
D) 43

Question 78

Evaluate the expression without using a calculator.
Use the formula $R = \log \left( \frac { \mathrm { a } } { \mathrm { T } } \right) + \mathrm { B }$ to find the intensity $\mathrm { R }$ on the Richter scale, given that amplitude a is 304 micrometers, time T between waves is $3.8$ seconds, and B is 3.1. Round answer to one decimal place.

A) $1.9$
B) $7.4$
C) $7.5$
D) 5

Question 79

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
$\log _ { 10 } ( 10,000 x )$

A) $4 + \log _ { 10 } \mathrm { x }$
B) $40 + \log _ { 10 } \mathrm { x }$
C) $4 \log _ { 10 } x$
D) $4 \mathrm { x }$

Question 80

Evaluate the expression without using a calculator.
$\log 0.01$

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