Deck 11: Exponential and Logarithmic Functions

Question 1

Solve the equation. $$\left( \frac { 4 } { 5 } \right) ^ { x } = \frac { 625 } { 256 }$$

A)x = 6
B)x = -4
C)x = -6
D)x = -5
E)x = 4

Answer 1

x = -4

Question 2

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

A)
B)
C)
D)

Answer 2

Question 3

Suppose that the present population of a city is 550,000. Use the equation P ( t )= 550,000 e ^{0.034 t} to estimate future growth. Estimate the population 50 years from now.

A)P = 3,012,651
B)P = 3,010,671
C)P = 3,010,093
D)P = 3,010,805
E)P = 3,010,950

Answer 3

P = 3,010,671

Question 4

Complete the following chart, which illustrates what happens to $8,000 invested at 12% for different lengths of time and different numbers of compounding periods. Round all of your answers to the nearest dollar. $$\begin{array} { | c | c | c | c | c | } \hline & \text { l year } & \text { T years } & \text { 14 years } & \text { 28 years } \\\hline \text { Compounded annually } & & & & \\\hline \text { Compoundedsemiannually } & & & & \\\hline \text { Compounded quatterly } & & & & \\\hline \text { Compounded monthly } & & & & \\\hline \text { Compounded continuously } & & & & \\\hline\end{array}$$

A) $\begin{array} { | c | c | c | c | } \hline 4480 & 8843 & 19548 & 95535 \\\hline 4494 & 9044 & 20447 & 104517 \\\hline 4502 & 9152 & 20938 & 109605 \\\hline 4507 & 9227 & 21284 & 113251 \\\hline 4510 & 9265 & 21462 & 115157 \\\hline\end{array}$
B) $\begin{array} { | l | l | l | l | } \hline 9020 & 18531 & 42924 & 230314 \\\hline 9015 & 18454 & 42568 & 226502 \\\hline 9004 & 18303 & 41877 & 219209 \\\hline 8989 & 18087 & 40893 & 209035 \\\hline 8960 & 17685 & 39097 & 191071 \\\hline\end{array}$
C) $\begin{array} { | c | c | c | c | } \hline 8960 & 17685 & 39097 & 191071 \\\hline 26966 & 54262 & 122680 & 627104 \\\hline 27012 & 54910 & 125631 & 657628 \\\hline 27044 & 55361 & 127703 & 679505 \\\hline 27060 & 55593 & 128773 & 690941 \\\hline\end{array}$
D) $\begin{array} { | c | c | c | c | } \hline 8960 & 17685 & 39097 & 191071 \\\hline 8989 & 18087 & 40893 & 209035 \\\hline 9004 & 18303 & 219209 & 41877 \\\hline 9015 & 18454 & 226502 & 42568 \\\hline 9020 & 18531 & 230314 & 230314 \\\hline\end{array}$
E) $\begin{array} { | l | l | l | l | } \hline 8960 & 17685 & 39097 & 191071 \\\hline 8989 & 18087 & 40893 & 209035 \\\hline 9004 & 18303 & 41877 & 219209 \\\hline 9015 & 18454 & 42568 & 226502 \\\hline 9020 & 18531 & 42924 & 230314 \\\hline\end{array}$

Question 5

Complete the following chart, which illustrates what happens to $16,000 invested at various rates of interest for different lengths of time but always compounded continuously. Round your answers to the nearest dollar. $$\begin{array} { | c | c | c | c | c | } \hline & \mathbf { 8 } \% & \mathbf { 1 0 } \% & \mathbf { 1 2 } \% & \mathbf { 1 4 } \% \\\hline 5 \text { years } & & & & \\\hline 8 \text { years } & & & & \\\hline 11 \text { years } & & & & \\\hline 14 \text { years } & & & & \\\hline 17 \text { years } & & & & \\\hline\end{array}$$

A) $\begin{array} { | c | c | c | c | c | } \hline & \mathbf { 8 } \% & \mathbf { 1 0 } \% & \mathbf { 1 2 } \% & \mathbf { 1 4 } \% \\\hline 5 \text { years } & 23869 & 26380 & 29154 & 32220 \\\hline 8 \text { years } & 30344 & 35609 & 41787 & 49038 \\\hline 11 \text { years } & 38574 & 48067 & 59895 & 74633 \\\hline 14 \text { years } & 49038 & 64883 & 85849 & 113589 \\\hline 17 \text { years } & 62339 & 87583 & 123050 & 172878 \\\hline\end{array}$
B) $\begin{array} { | c | c | c | c | c | } \hline & \mathbf { 8 } \% & \mathbf { 1 0 } \% & \mathbf { 1 2 } \% & \mathbf { 1 4 } \% \\\hline 5 \text { years } & 32220 & 29154 & 26380 & 23869 \\\hline 8 \text { years } & 49038 & 41787 & 35609 & 30344 \\\hline 11 \text { years } & 74633 & 59895 & 48067 & 38574 \\\hline 14 \text { years } & 113589 & 85849 & 64883 & 49038 \\\hline 17 \text { years } & 172878 & 123050 & 87583 & 62339 \\\hline\end{array}$
C) $\begin{array} { | c | c | c | c | c | } \hline & \mathbf { 8 } \% & \mathbf { 1 0 } \% & \mathbf { 1 2 } \% & \mathbf { 1 4 } \% \\\hline 5 \text { years } & 47738 & 52759 & 58308 & 64440 \\\hline 8 \text { years } & 60687 & 71217 & 83574 & 98075 \\\hline 11 \text { years } & 77149 & 96133 & 119789 & 149267 \\\hline 14 \text { years } & 98075 & 129766 & 171698 & 227178 \\\hline 17 \text { years } & 124678 & 175166 & 246099 & 345757 \\\hline\end{array}$
D) $\begin{array} { | c | c | c | c | c | } \hline & \mathbf { 8 } \% & \mathbf { 1 0 } \% & \mathbf { 1 2 } \% & \mathbf { 1 4 } \% \\\hline 5 \text { years } & 11935 & 13190 & 14577 & 16110 \\\hline 8 \text { years } & 15172 & 17804 & 20894 & 24519 \\\hline 11 \text { years } & 19287 & 24033 & 29947 & 37317 \\\hline 14 \text { years } & 24519 & 32442 & 42924 & 56795 \\\hline 17 \text { years } & 31170 & 43792 & 61525 & 86439 \\\hline\end{array}$
E) $\begin{array} { | c | c | c | c | c | } \hline & \mathbf { 8 } \% & \mathbf { 1 0 } \% & \mathbf { 1 2 } \% & \mathbf { 1 4 } \% \\\hline 5 \text { years } & 62339 & 87583 & 123050 & 172878 \\\hline 8 \text { years } & 49038 & 64883 & 85849 & 113589 \\\hline 11 \text { years } & 38574 & 48067 & 59895 & 74633 \\\hline 14 \text { years } & 30344 & 35609 & 41787 & 49038 \\\hline 17 \text { years } & 23869 & 26380 & 29154 & 32220 \\\hline\end{array}$

Question 6

Suppose that Nora invested $400 at 8.5% compounded annually for 6 years and Patti invested $400 at 8% compounded quarterly for 6 years. At the end of 6 years, who will have the most money and by how much (to the nearest dollar)?

A)Patti by $9
B)Nora by $1
C)Patti by $1
D)Nora by $9
E)They will have equal amounts of money.

Question 7

Suppose that a certain radioactive substance has a half-life of 45 years. If there are presently 1,920 milligrams of the substance, how much, to the nearest milligram, will remain after 180 years?

A)Q = 130 milligram
B)Q = 120 milligram
C)Q = 135 milligram
D)Q = 95 milligram
E)Q = 125 milligram

Question 8

Solve the equation. $$\left( \frac { 1 } { 5 } \right) ^ { 2 x } = 625$$

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

Question 9

Graph the exponential function. f ( x )= 3 ^x

A)
B)
C)
D)

Question 10

Graph the exponential function. f ( x )= 3 ^{x + 2}

A)
B)
C)
D)

Question 11

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

A)
B)
C)
D)
E)none of these

Question 12

The number of bacteria present at a given time under certain conditions is given by the equation Q = 4,000 e ^{0.01 t} , where t is expressed in minutes. How many bacteria are present at the end of 45 minutes? Please round the answers to the nearest whole.

A)Q = 6,344 bacteria
B)Q = 6,273 bacteria
C)Q = 6,287 bacteria
D)Q = 6,298 bacteria
E)Q = 6,282 bacteria

Question 13

The half-life of radium is approximately 1,600 years. If the present amount of radium in a certain location is 900 grams, how much will remain after 400 years? Express your answer to the nearest gram.

A)Q = 734 gram
B)Q = 768 gram
C)Q = 757 gram
D)Q = 752 gram
E)Q = 744 gram

Question 14

The number of grams Q of a certain radioactive substance present after t seconds is given by the equation $Q = 7,000 e ^ { - 0.1 t }$ . How many grams remain after 17 seconds? Please round the answers to the nearest whole.

A)Q = 1,355 grams
B)Q = 1,329 grams
C)Q = 1,536 grams
D)Q = 1,278 grams
E)Q = 1,279 grams

Question 15

Graph the exponential function. f ( x )= e ^x - 5

A)
B)
C)
D)

Question 16

Solve the equation. $$\left( 4 ^ { 3 x - 6 } \right) \left( 4 ^ { x + 4 } \right) = 64$$

A) $x = - \frac { 3 } { 4 }$
B) $x = - \frac { 5 } { 4 }$
C) $x = \frac { 5 } { 4 }$
D) $x = \frac { 1 } { 4 }$
E) $x = \frac { 9 } { 4 }$

Question 17

Write 2 ⁹ = 512 in logarithmic form. For example, 2 ³ = 8 becomes log ₂ 8 = 3 in logarithmic form.

A)log ₂ 9 = 512
B)log ₅₁₂ 2 = 9
C)log ₅₁₂ 9 = 2
D)log ₂ 512 = 9
E)log ₉ 512 = 2

Question 18

Suppose that in a certain culture, the equation $\mathcal { Q } ( t ) = 900 e ^ { 0.5 t }$ 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)3,670 bacteria
B)6,605 bacteria
C)1,631 bacteria
D)906 bacteria
E)2,446 bacteria

Question 19

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

A)
B)
C)
D)

Question 20

Write 10 ⁶ = 1,000,000 in logarithmic form. For example, 2 ³ = 8 becomes log ₂ 8 = 3 in logarithmic form.

A)log ₁₀ 6 = 1,000,000
B)log ₆ 1,000,000 = 10
C)log _1,000,000 10 = 6
D)log _1,000,000 6 = 10
E)log ₁₀ 1,000,000 = 6

Question 21

Solve the equation log ₉ x + log ₉ ( x - 24)= 2.

A)x = 29
B)x = -3
C)x = 27
D)x = 3
E)x = 729

Question 22

Use your calculator to find x when given ln x . Express answer to four decimal places. $$\ln x = 0.5973$$

A)1.8672
B)1.8072
C)1.8472
D)1.8172
E)1.8372

Question 23

Solve the equation log ₆ x = 2.

A)x = 64
B)x = 42
C)x = 2
D)x = 6
E)x = 36

Question 24

Graph the function $y = 3 + \log _ { 2 } x$ . Remember that the graph of $f ( x ) = \log _ { 2 } x$ is given.

A)
B)
C)

Question 25

Evaluate the following. $$\log _ { 8 } 8 ^ { 6 }$$

A)8
B)1
C)6
D)0
E)none of these

Question 26

Write the following equation in logarithmic form. For example, 2 ³ = 8 becomes log ₂ 8 = 3 in logarithmic form. $$2 ^ { - 5 } = \frac { 1 } { 32 }$$

A) $\log _ { 5 } \frac { 1 } { 32 } = - 2$
B) $\log _ { 2 } 5 = - 32$
C) $\log _ { 2 } \frac { 1 } { 5 } = - 32$
D) $\log _ { 2 } \frac { 1 } { 32 } = - 5$
E) $\log _ { 5 } \frac { 1 } { 2 } = - 2 ^ { 5 }$

Question 27

Solve the equation log ₃ x + log ₃2 = 2.

A)x = 18
B)x = 5.5
C)x = 4.5
D)x = 5
E)x = 4

Question 28

Solve the logarithmic equation. Check the solution. $$\log ( 8 x + 5 ) - \log ( 7 x + 8 ) = 0$$

A)x = 1
B)x = 0
C)x = 3
D)x = 4

Question 29

Solve the logarithmic equation. Check the solution. $$\log \left( x ^ { 2 } + 7 x \right) = \log \left( x ^ { 2 } + 56 \right)$$

A)x = 9
B)x = 8
C)x = 11
D)x = 10

Question 30

Graph $y = \log _ { 6 } x$ by reflecting the graph of $g ( x ) = 6 ^ { x }$ across the line y = x .

A)
B)
C)

Question 31

Graph $y = \log _ { \frac { 1 } { 4 } } x$ by graphing $\left( \frac { 1 } { 4 } \right) ^ { y } = x$ .

A)
B)
C)

Question 32

Solve the logarithmic equation. Check the solution. $$\log x ^ { 3 } = 9$$

A) $x =$ 1,000
B) $x =$ 100
C) $x =$ 10,000
D) $x =$ 10
E) $x =$ 1

Question 33

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $$\log _ { \frac { 1 } { 3 } } 3$$

A)0.5769
B)- 1.0000
C)1.1005
D)- 1.0078
E)1.7654

Question 34

Solve the equation. $$\log _ { \frac { 2 } { 3 } } \frac { n } { p } = 5$$

A) $\frac { n } { p } = 32$
B) $\frac { n } { p } = 243$
C) $\frac { n } { p } = \frac { 32 } { 243 }$
D) $\frac { n } { p } = \frac { 243 } { 32 }$
E)none of these

Question 35

Evaluate the following. $$\log _ { b } 1$$

A)0
B) $b$
C)1
D) $\frac { 1 } { b }$
E)none of these

Question 36

Assume that $x$ , $y$ , $Z$ , and $b ( b \neq 1 )$ are positive numbers Use the properties of logarithms to write the expression in terms of the logarithms of $x$ , $y$ , and $Z$ . $\log _ { b } x y ^ { 5 } z ^ { 6 }$

A) $\log _ { b } x + \frac { \log _ { b } y } { 5 } + \frac { \log _ { b } z } { 6 }$
B) $\log _ { b } x + \left( \log _ { b } y \right) ^ { 5 } + \left( \log _ { b } z \right) ^ { 6 }$
C) $\log _ { b } x + 5 \log _ { b } y + 6 \log _ { b } z$
D) $- \log _ { b } x - 5 \log _ { b } y - 6 \log _ { b } z$
E) $\log _ { b } x - \frac { \log _ { b } y } { 5 } - \frac { \log _ { b } z } { 6 }$

Question 37

Assume that $x$ , $y$ , $z$ , and $\text { a }$ $( b \neq 1 )$ are positive numbers.Use the properties of logarithms to write the expression in terms of the logarithms of x , y , and z . $\log _ { b } \frac { x } { y z }$

A)log _b x log _b y log _b z
B)log _b x + log _b y + log _b z
C)- log _b x - log _b y - log _b z
D)log _b x - log _b y - log _b z

Question 38

Solve the equation. $$\log _ { 27 } x = \frac { 4 } { 3 }$$

A)x = 2
B)x = 68
C)x = 16
D)x = 81
E)x = 64

Question 39

Evaluate the expression log ₉ (log ₂ 8).

A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 3 }$
C) $9$
D) $2$
E) $\frac { 1 } { 2 }$

Question 40

Solve the equation log ₃ (2 x - 1)- log ₃ ( x - 2)= 1.

A)x = 4
B)x = 5
C)x = 2
D)x = 7
E)x = 6

Question 41

Find the value of $x$ . $\log _ { 81 } x = \frac { 1 } { 2 }$

A)9
B) $\frac { 1 } { 9 }$
C)2
D) $\frac { 1 } { 2 }$
E) $1$

Question 42

Find the value of $x$ . $\log _ { 3 } 81 = x$

A)3
B)5
C)6
D)2
E)4

Question 43

How long will it take $700 to double itself if it is invested at 13% interest compounded semiannually? Please round the answer to the nearest tenth.

A)5.5 years
B)8.5 years
C)6.0 years
D)4.5 years
E)6.5 years

Question 44

Solve the exponential equation and express solution to the nearest hundredth. $$2 e ^ { x } = 28.8$$

A)x = 2.37
B)x = 2.97
C)x = 2.47
D)x = 2.67
E)x = 2.57

Question 45

A principal sum of money P is invested at the end of each year in an annuity earning annual interest at a rate of r . The amount in the annuity account will be A dollars after n years, where $n = \frac { \log \left( \frac { A v } { P } + 1 \right) } { \log ( 1 + r ) }$ If $2,000 is invested each year in an annuity earning 9% annual interest, how long will it take for the account to be worth $15,000? Round the answer to the nearest tenth of the year.

A)7.5 years
B)6.0 years
C)4.2 years
D)6.7 years
E)5.1 years

Question 46

Use your calculator to find x when given ln x . Express answer to four decimal places. $$\ln x = 2.5138$$

A)12.4018
B)12.3418
C)12.3818
D)12.3518
E)12.3718

Question 47

Solve the exponential equation and express approximate solutions to the nearest hundredth. e ^{x - 4} = 38

A)x = 7.11
B)x = 4.00
C)x = 8.17
D)x = 7.64
E)x = 7.52

Question 48

Use a calculator and the change-of-base formula to find the logarithm to four decimal places. $$\log _ { 3 } \pi$$

A)0.6189
B)1.0420
C)1.8074
D)1.0342
E)0.9415

Question 49

Solve the equation. $$10 ^ { x } = 0.01$$

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

Question 50

Use your calculator to find x when given ln x . Express answer to four decimal places. $$\ln x = - 2.2449$$

A)0.1059
B)0.1259
C)0.1559
D)0.1359
E)0.0959

Question 51

Use a calculator and the change-of-base formula to find the logarithm to four decimal places.

Question 52

Calculate how many times more intense an earthquake with a Richter number of 8.3 is than an earthquake with a Richter number of 6.9. Please round the answer to the whole number.

A)30
B)33
C)25
D)35
E)27

Question 53

An earthquake in the Japan in 2011 was about 10 9 times as intense as the reference intensity. Find the Richter number for that earthquake. NOTE: Seismologists use the Richter scale to measure and report the magnitude of earthquakes. The equation $R = \log \frac { I } { I _ { 0 } }$ compares the intensity I of an earthquake to a minimum or reference intensity I 0.

A)10⁹
B)10
C)9
D)none of these

Question 54

Business equipment is often depreciated using the double declining-balance method. In this method, a piece of equipment with a life expectancy of N years, costing C dollars, will depreciate to a value of V dollars in n years, where n is given by the formula: $n = \frac { \log V - \log C } { \log \left( 1 - \frac { 2 } { N } \right) }$ A computer that cost $10,000 has a life expectancy of 5 years. If it has depreciated to a value of $2,000, how old is it? Round the answer to the nearest tenth of the year.

A)3.2 years old
B)2.2 years old
C)1.9 years old
D)2.5 years old
E)2.8 years old

Question 55

An earthquake has a period of $\frac { 1 } { 2 }$ second, and an amplitude of $500,000$ micrometers. Find its measure on the Richter scale.

A) $3$
B) $2$
C) $5$
D) $6$
E) $8$

Question 56

Solve the logarithmic equation. Check the solution. $$\log _ { 2 } x = \log _ { 2 } \frac { 1 } { x } + 6$$

A) $x =$ 10
B) $x =$ 27
C) $x =$ 6.
D) $x =$ 6
E) $x =$ 8

Question 57

Use the change of base formula to identify the expression that is equivalent to $\log _ { 9 } 7$ .

A) $\frac { \log 7 } { \log 9 }$
B) $\frac { \log 9 } { \log 7 }$
C) $\frac { \ln 9 } { \ln 7 }$
D) $9 \log 7$