Deck 5: Inverse, Exponential, and Logarithmic Functions

Question 1

Match each equation with its graph.

- $y=\ln x$

A)
B)
C)
D)

Answer 1

Question 2

Match each equation with its graph.

- $y=\log _{1 / 4} x$

A)
B)
C)
D)

Answer 2

Question 3

Match each equation with its graph.

- $y=\left(\frac{1}{4}\right)^{x}$

A)
B)
C)
D)

Answer 3

Question 4

Match each equation with its graph.

- $y=e^{x}$

A)
B)
C)
D)

Question 5

Consider the function $f(x)=-5^{x-4}+2$ .
(a) Graph it in the standard viewing window of your calculator.
(b) Give the domain and range of $f$ .
(c) Does the graph have an asymptote? If so, is it vertical or horizontal, and what is its equation?
(d) Find the $x$ - and $y$ -intercepts analytically and use the graph from part (a) to support your answers graphically.
(e) Find $f^{-1}(x)$ .

Question 6

Solve the equation $16^{2 x-6}=\left(\frac{1}{8}\right)^{4 x-2}$ analytically.

Question 7

Suppose that $\$ 8100$ is invested at 3.8 % for 20 years. Find the total amount present at the end of this time period if the interest is compounded (a) quarterly and (b) continuously.

Question 8

One of your friends is taking another mathematics course and tells you, "I have no idea what an expression like $\log _{7} 12$ really means." Write an explanation of what it means, and tell how you can find an approximation for it with a calculator.

Question 9

Use a calculator to find an approximation of each logarithm to the nearest thousandth.
(a) $\log _{2} 4.16$
(b) $\log 37.6$
(c) $\ln 639$

Question 10

Use the power, quotient, and product properties of logarithms to write $\ln \frac{\sqrt[3]{x^{2} y}}{z^{4}}$ as an equivalent expression.

Question 11

Solve $A=P e^{r t}$ for $r$ .

Question 12

Consider the equation $\log _{2} x+\log _{2}(x-4)=5$ .
(a) Solve the equation analytically. If there is an extraneous value, what is it?
(b) To support the solution in part (a), we may graph $y_{1}=\log _{2} x+\log _{2}(x-4)-5$ and find the $x$ -intercept.
Write an expression for $y_{1}$ using the change-of-base rule with base 10 , and graph the function to support the solution from part (a).
(c) Use the graph to solve the inequality $\log _{2} x+\log _{2}(x-4)<5$ .

Question 13

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $2 e^{3 x-1}=10$

Question 14

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $10^{1-x}=3^{2 x+1}$

Question 15

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $\ln (\log x)=1$

Question 16

An unstable radioactive isotope decays according to the equation $y=2.53(.97)^{t}$ where $y$ is the number of grams remaining and $t$ is the time measured in minutes. Match each question with one of the solutions A, B, C, or D.

-How long will it take for the material to decay to 2 grams?

A) Evaluate $2.53(.97)^{1 / 2}$
B) Evaluate $2.53(.97)^{2}$
C) Solve $2=2.53(.97)^{t}$
D) Solve $\frac{1}{2}(2.53)=2.53(.97)^{t}$

Question 17

An unstable radioactive isotope decays according to the equation $y=2.53(.97)^{t}$ where $y$ is the number of grams remaining and $t$ is the time measured in minutes. Match each question with one of the solutions A, B, C, or D.

-How many grams are remaining after 2 minutes?

A) Evaluate $2.53(.97)^{1 / 2}$
B) Evaluate $2.53(.97)^{2}$
C) Solve $2=2.53(.97)^{t}$
D) Solve $\frac{1}{2}(2.53)=2.53(.97)^{t}$

Question 18

An unstable radioactive isotope decays according to the equation $y=2.53(.97)^{t}$ where $y$ is the number of grams remaining and $t$ is the time measured in minutes. Match each question with one of the solutions A, B, C, or D.

-How many grams are remaining after 30 seconds?

A) Evaluate $2.53(.97)^{1 / 2}$
B) Evaluate $2.53(.97)^{2}$
C) Solve $2=2.53(.97)^{t}$
D) Solve $\frac{1}{2}(2.53)=2.53(.97)^{t}$

Question 19

An unstable radioactive isotope decays according to the equation $y=2.53(.97)^{t}$ where $y$ is the number of grams remaining and $t$ is the time measured in minutes. Match each question with one of the solutions A, B, C, or D.

-How long will it take for the material to decay to half its initial amount?

A) Evaluate $2.53(.97)^{1 / 2}$
B) Evaluate $2.53(.97)^{2}$
C) Solve $2=2.53(.97)^{t}$
D) Solve $\frac{1}{2}(2.53)=2.53(.97)^{t}$

Question 20

When a loaf of bread is removed from the freezer to thaw, the temperature of the bread increases. Graph each of the following functions on the interval $[0,50]$ . Use $[0,100]$ for the range of $A(t)$ . Use a graphing calculator to determine the function that best describes the temperature of the bread $A(t)$ (in degrees Fahrenheit) $t$ minutes after it is removed from the freezer, if the initial temperature of the bread was 30 degrees Fahrenheit.
(a) $A(t)=-.2 t^{2}+4 t+30$
(b) $A(t)=70-40 e^{.01 t}$
(c) $A(t)=30 \ln (t+1)$
(d) $A(t)=70-40 e^{-.06 t}$

Question 21

A sample of radioactive material has a half-life of about 1600 years. An initial sample weighs 18 grams.
(a) Find a formula for the decay function for this material.
(b) Find the amount left after 6400 years
(c) Find the time for the initial amount to decay to 4.5 grams.

Question 22

Match each equation with its graph

- $y=\ln x$

A)
B)
C)
D)

Question 23

Match each equation with its graph

- $y=e^{x}$

A)
B)
C)
D)

Question 24

Match each equation with its graph

- $y=\log _{1 / 10} x$

A)
B)
C)
D)

Question 25

Match each equation with its graph

- $y=\left(\frac{1}{10}\right)^{x}$

A)
B)
C)
D)

Question 26

Consider the function $f(x)=-2^{x-3}-1$ .
(a) Graph it in the standard viewing window of your calculator.
(b) Give the domain and range of $f$ .
(c) Does the graph have an asymptote? If so, is it vertical or horizontal, and what is its equation?
(d) Find the $x$ - and $y$ -intercepts analytically and use the graph from part (a) to support your answers graphically.
(e) Find $f^{-1}(x)$ .

Question 27

Solve the equation $9^{x+3}=\left(\frac{1}{3}\right)^{x-7}$ analytically.

Question 28

Suppose that $\$ 12,000$ is invested at $5.8 \%$ for 6 years. Find the total amount present at the end of this time period if the interest is compounded (a) monthly and (b) continuously.

Question 29

One of your friends is taking another mathematics course and tells you "I know that the expression $\log _{2}(-4)$ is undefined because the definition says that for any expression of the form $\log _{a} x, x$ can't be 0 and it can't be negative, but I don't understand why this is true." Write an explanation for your friend.

Question 30

Use a calculator to find an approximation of each logarithm to the nearest thousandth.
(a) $\ln 21.6$
(b) $\log _{3} 19.1$
(c) $\log 153$

Question 31

Use the power, quotient, and product properties of logarithms to write $\log \frac{a}{\sqrt{b c^{3}}}$ as an equivalent expression.

Question 32

Solve $d=10 \log \frac{I}{I_{0}}$ for $I$ .

Question 33

Consider the equation $\log _{6} x+\log _{6}(x-5)=1$ .
(a) Solve the equation analytically. If there is an extraneous value, what is it?
(b) To support the solution in part (a), we may graph $y_{1}=\log _{6} x+\log _{6}(x-5)-1$ and find the $x$ -intercept.
Write an expression for $y_{1}$ using the change-of-base rule with base 10, and graph the function to support the solution from part (a).
(c) Use the graph to solve the inequality $\log _{6} x+\log _{6}(x-5)<1$ .

Question 34

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $4 e^{5 x+1}=8$

Question 35

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $5^{2 x+1}=3^{4 x-1}$

Question 36

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $\log (\ln 3 x)=1$

Question 37

A water tank has been contaminated with salt and must be flushed with pure water. The concentration of salt in the tank is given by the equation $y=.1+3.02 e^{-.4 t}$ , where $y$ is the concentration of salt in milligrams per gallon and $t$ is the time, in hours, since flushing began. Match each question with one of the solutions $\mathrm{A}, \mathrm{B}, \mathrm{C}$ , or $\mathrm{D}$ .

-What is the concentration of salt after 2 hours?

A) Evaluate $.1+3.02 e^{-.4(1 / 2)}$
B) Solve $\frac{1}{2}(3.12)=.1+3.02 e^{-.4 t}$
C) Evaluate $.1+3.02 e^{-.4(2)}$
D) Solve $2=.1+3.02 e^{-.4 t}$

Question 38

A water tank has been contaminated with salt and must be flushed with pure water. The concentration of salt in the tank is given by the equation $y=.1+3.02 e^{-.4 t}$ , where $y$ is the concentration of salt in milligrams per gallon and $t$ is the time, in hours, since flushing began. Match each question with one of the solutions $\mathrm{A}, \mathrm{B}, \mathrm{C}$ , or $\mathrm{D}$ .

-How long will it take for the concentration to reach half its initial value?

A) Evaluate $.1+3.02 e^{-.4(1 / 2)}$
B) Solve $\frac{1}{2}(3.12)=.1+3.02 e^{-.4 t}$
C) Evaluate $.1+3.02 e^{-.4(2)}$
D) Solve $2=.1+3.02 e^{-.4 t}$

Question 39

A water tank has been contaminated with salt and must be flushed with pure water. The concentration of salt in the tank is given by the equation $y=.1+3.02 e^{-.4 t}$ , where $y$ is the concentration of salt in milligrams per gallon and $t$ is the time, in hours, since flushing began. Match each question with one of the solutions $\mathrm{A}, \mathrm{B}, \mathrm{C}$ , or $\mathrm{D}$ .

-How long will it take for the concentration to reach $2 \mathrm{mg} / \mathrm{gal}$ ?

A) Evaluate $.1+3.02 e^{-.4(1 / 2)}$
B) Solve $\frac{1}{2}(3.12)=.1+3.02 e^{-.4 t}$
C) Evaluate $.1+3.02 e^{-.4(2)}$
D) Solve $2=.1+3.02 e^{-.4 t}$

Question 40

A water tank has been contaminated with salt and must be flushed with pure water. The concentration of salt in the tank is given by the equation $y=.1+3.02 e^{-.4 t}$ , where $y$ is the concentration of salt in milligrams per gallon and $t$ is the time, in hours, since flushing began. Match each question with one of the solutions $\mathrm{A}, \mathrm{B}, \mathrm{C}$ , or $\mathrm{D}$ .

-What is the concentration of salt after 30 minutes?

A) Evaluate $.1+3.02 e^{-.4(1 / 2)}$
B) Solve $\frac{1}{2}(3.12)=.1+3.02 e^{-.4 t}$
C) Evaluate $.1+3.02 e^{-.4(2)}$
D) Solve $2=.1+3.02 e^{-.4 t}$

Question 41

A kiln is used to heat materials to high temperatures. Graph each of the following functions on the interval $[0,10]$ . Use $[0,450]$ for the range of $A(t)$ . Use a graphing calculator to determine the function that best describes the temperature of the material $A(t)$ (in degrees Fahrenheit) $t$ minutes after it is placed in the kiln, if the initial temperature of the material was 50 degrees Fahrenheit.
(a) $A(t)=550-100 e^{.1 t}$
(b) $A(t)=550 \log (t-1)$
(c) $A(t)=550-500 e^{-.15 t}$
(d) $A(t)=t^{2}-40 t+550$

Question 42

A sample of radioactive material has a half-life of about 1200 years. An initial sample weighs 20 grams.
(a) Find a formula for the decay function for this material.
(b) Find the amount left after 6000 years.
(c) Find the time for the initial amount to decay to 2.5 grams.

Question 43

Match each equation with its graph.

- $y=e^{x}$

A)
B)
C)
D)

Question 44

Match each equation with its graph.

- $y=\left(\frac{1}{3}\right)^{x}$

A)
B)
C)
D)

Question 45

Match each equation with its graph.

- $y=\ln x$

A)
B)
C)
D)

Question 46

Match each equation with its graph.

- $y=\log _{1 / 3} x$

A)
B)
C)
D)

Question 47

Consider the function $f(x)=-3^{x+2}-1$ .
(a) Graph it in the standard viewing window of your calculator.
(b) Give the domain and range of $f$ .
(c) Does the graph have an asymptote? If so, is it vertical or horizontal, and what is its equation?
(d) Find the $x$ - and $y$ -intercepts analytically and use the graph from part (a) to support your answers graphically.
(e) Find $f^{-1}(x)$ .

Question 48

Solve the equation $\left(\frac{1}{36}\right)^{2 x-3}=6^{3 x+1}$ analytically.

Question 49

Suppose that $\$ 5000$ is invested at $4.3 \%$ for 15 years. Find the total amount present at the end of this time period if the interest is compounded (a) semiannually and (b) continuously.

Question 50

One of your friends is taking another mathematics course and tells you, "I have no idea what an expression like $\log _{6} 27$ really means." Write an explanation of what it means, and tell how you can find an approximation for it with a calculator.

Question 51

Use a calculator to find an approximation of each logarithm to the nearest thousandth.
(a) $\log 17.6$
(b) $\log _{9} 750$
(c) $\ln 901$

Question 52

Use the power, quotient, and product properties of logarithms to write $\log \frac{p^{3} q^{2}}{\sqrt[4]{r}}$ as an equivalent expression.

Question 53

Solve $T=T_{0}+C e^{-k t}$ for $t$ .

Question 54

Consider the equation $\log _{4} x+\log _{4}(x+6)=2$ .
(a) Solve the equation analytically. If there is an extraneous value, what is it?
(b) To support the solution in part (a), we may graph $y_{1}=\log _{4} x+\log _{4}(x+6)-2$ and find the $x$ -intercept.
Write an expression for $y_{1}$ using the change-of-base rule with base 10 , and graph the function to support the solution from part (a).
(c) Use the graph to solve the inequality $\log _{4} x+\log _{4}(x+6)<2$ .

Question 55

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $3 e^{5 x-2}=12$

Question 56

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $6^{3-x}=3^{2 x-1}$

Question 57

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $\log (\ln x)=0$

Question 58

The concentration of pollutants in a stream is given by $y=.06 e^{-3 x}$ , where $y$ is the amount of pollutant in grams per liter and $x$ is the distance, in kilometers, downstream from the source of the pollution. Match each question with one of the solutions A, B, C, or D.

-How far downstream is the pollutant level equal to $.02 \mathrm{gm} / \mathrm{L}$ ?

A) Solve $.02=.06 e^{-3 x}$
B) Solve $\frac{1}{2}(.06)=.06 e^{-3 x}$
C) Evaluate $.06 e^{-3(.02)}$
D) Evaluate $.06 e^{-3(1 / 2)}$

Question 59

The concentration of pollutants in a stream is given by $y=.06 e^{-3 x}$ , where $y$ is the amount of pollutant in grams per liter and $x$ is the distance, in kilometers, downstream from the source of the pollution. Match each question with one of the solutions A, B, C, or D.

-What is the pollutant level .02 kilometer from the source?

A) Solve $.02=.06 e^{-3 x}$
B) Solve $\frac{1}{2}(.06)=.06 e^{-3 x}$
C) Evaluate $.06 e^{-3(.02)}$
D) Evaluate $.06 e^{-3(1 / 2)}$

Question 60

The concentration of pollutants in a stream is given by $y=.06 e^{-3 x}$ , where $y$ is the amount of pollutant in grams per liter and $x$ is the distance, in kilometers, downstream from the source of the pollution. Match each question with one of the solutions A, B, C, or D.

-What is the pollutant level 500 meters from the source?

A) Solve $.02=.06 e^{-3 x}$
B) Solve $\frac{1}{2}(.06)=.06 e^{-3 x}$
C) Evaluate $.06 e^{-3(.02)}$
D) Evaluate $.06 e^{-3(1 / 2)}$

Question 61

The concentration of pollutants in a stream is given by $y=.06 e^{-3 x}$ , where $y$ is the amount of pollutant in grams per liter and $x$ is the distance, in kilometers, downstream from the source of the pollution. Match each question with one of the solutions A, B, C, or D.

-How far downstream is the pollutant level half the amount at the source?

A) Solve $.02=.06 e^{-3 x}$
B) Solve $\frac{1}{2}(.06)=.06 e^{-3 x}$
C) Evaluate $.06 e^{-3(.02)}$
D) Evaluate $.06 e^{-3(1 / 2)}$

Question 62

When leftover food is placed in a refrigerator following a meal, the temperature of the food decreases. Graph each of the following functions on the interval $[0,50]$ . Use $[0,100]$ for the range of $A(t)$ . Use a graphing calculator to determine the function that best describes the temperature of the food $A(t)$ (in degrees Fahrenheit) $t$ minutes after it is placed in the refrigerator, if the initial temperature of the food was 90 degrees Fahrenheit.
(a) $A(t)=-.1 t^{2}+2 t+90$
(b) $A(t)=35+55 e^{-.04 t}$
(c) $A(t)=90 \ln (.05 t+1)$
(d) $A(t)=35+e^{.09 t}$

Question 63

A sample of radioactive material has a half-life of about 6000 years. An initial sample weighs 32 grams.
(a) Find a formula for the decay function for this material.
(b) Find the amount left after 30,000 years.
(c) Find the time for the initial amount to decay to .5 gram.

Question 64

Match each equation with its graph.

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

A)
B)
C)
D)

Question 65

Match each equation with its graph.

- $y=\log _{1 / 2} x$

A)
B)
C)
D)

Question 66

Match each equation with its graph.

- $y=e^{x}$

A)
B)
C)
D)

Question 67

Match each equation with its graph.

- $y=\ln x$

A)
B)
C)
D)

Question 68

Consider the function $f(x)=-4^{x+1}+3$ .
(a) Graph it in the standard viewing window of your calculator.
(b) Give the domain and range of $f$ .
(c) Does the graph have an asymptote? If so, is it vertical or horizontal, and what is its equation?
(d) Find the $x$ - and $y$ -intercepts analytically and use the graph from part (a) to support your answers graphically.
(e) Find $f^{-1}(x)$ .

Question 69

Solve the equation $25^{x-9}=\left(\frac{1}{125}\right)^{2 x-2}$ analytically.

Question 70

Suppose that $\$ 25,000$ is invested at $5.7 \%$ for 12 years. Find the total amount present at the end of thime period if the interest is compounded (a) quarterly and (b) continuously.

Question 71

One of your friends is taking another mathematics course and tells you "I know that the expression $\log _{3} 0$ is undefined because the definition says that for any expression of the form $\log _{a} x, x$ can't be 0 and it can't be negative, but I don't understand why this is true."' Write an explanation for your friend.

Question 72

Use a calculator to find an approximation of each logarithm to the nearest thousandth.
(a) $\ln 21.7$
(b) $\log 83.5$
(c) $\log _{4} 752.4$

Question 73

Use the power, quotient, and product properties of logarithms to write $\ln \frac{\sqrt[5]{w}}{x^{2} y^{9}}$ as an equivalent expression.

Question 74

Solve $Q=Q_{0} e^{k t}$ for $t$ .

Question 75

Consider the equation $\log _{3} x+\log _{3}(x-8)=2$ .
(a) Solve the equation analytically. If there is an extraneous value, what is it?
(b) To support the solution in part (a), we may graph $y_{1}=\log _{3} x+\log _{3}(x-8)-2$ and find the $x$ -intercept.
Write an expression for $y_{1}$ using the change-of-base rule with base 10 , and graph the function to support the solution from part (a).
(c) Use the graph to solve the inequality $\log _{3} x+\log _{3}(x-8)<2$ .

Question 76

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $2 e^{7 x+3}=6$

Question 77

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $12^{2 x-2}=6^{2 x-1}$

Question 78

solve each equation. Give the solution set (a) with an exact value and then (b) with an approximation to the nearest thousandth.
- $\ln (\log 3 x)=-1$

Question 79

The atmospheric pressure at a given altitude is given by $y=14.7 e^{-.0000385 x}$ , where $y$ is the atmospheric pressure in pounds per square inch and $x$ is the altitude, in feet. Match each question with one of the solutions A, B, C, or D.

-What is the atmospheric pressure at an altitude of 10 feet?

A) Solve $\frac{1}{10}(14.7)=14.7 e^{-.0000385 x}$
B) Evaluate $14.7 e^{-.0000385(3 \cdot 10)}$
C) Solve $10=14.7 e^{-.0000385 x}$
D) Evaluate $14.7 e^{-.0000385(10)}$

Question 80

The atmospheric pressure at a given altitude is given by $y=14.7 e^{-.0000385 x}$ , where $y$ is the atmospheric pressure in pounds per square inch and $x$ is the altitude, in feet. Match each question with one of the solutions A, B, C, or D.

-What is the atmospheric pressure at an altitude of 10 yards?

A) Solve $\frac{1}{10}(14.7)=14.7 e^{-.0000385 x}$
B) Evaluate $14.7 e^{-.0000385(3 \cdot 10)}$
C) Solve $10=14.7 e^{-.0000385 x}$
D) Evaluate $14.7 e^{-.0000385(10)}$