# Chapter 10 Review Exercises

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## Chapter Review Exercises

### Finding Composite and Inverse Functions

##### Exercise $$\PageIndex{1}$$ Find and Evaluate Composite Functions

In the following exercises, for each pair of functions, find

1. $$(f \circ g)(x)$$
2. $$(g \circ f)(x)$$
3. $$(f \cdot g)(x)$$

1. $$f(x)=7 x-2$$ and $$g(x)=5 x+1$$

2. $$f(x)=4 x$$ and $$g(x)=x^{2}+3 x$$

2.

1. $$4 x^{2}+12 x$$
2. $$16 x^{2}+12 x$$
3. $$4 x^{3}+12 x^{2}$$
##### Exercise $$\PageIndex{2}$$ Find and Evaluate Composite Functions

In the following exercises, evaluate the composition.

1. For functions $$f(x)=3 x^{2}+2$$ and $$g(x)=4 x-3$$, find
1. $$(f \circ g)(-3)$$
2. $$(g \circ f)(-2)$$
3. $$(f \circ f)(-1)$$
2. For functions $$f(x)=2 x^{3}+5$$ and $$g(x)=3 x^{2}-7$$, find
1. $$(f \circ g)(-1)$$
2. $$(g \circ f)(-2)$$
3. $$(g \circ g)(1)$$

2.

1. $$-123$$
2. $$356$$
3. $$41$$
##### Exercise $$\PageIndex{3}$$ Determine Whether a Function is One-to-One

In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

1. $$\begin{array}{l}{\{(-3,-5),(-2,-4),(-1,-3),(0,-2)} , {(-1,-1),(-2,0),(-3,1) \}}\end{array}$$
2. $$\begin{array}{l}{\{(-3,0),(-2,-2),(-1,0),(0,1)} , {(1,2),(2,1),(3,-1) \}}\end{array}$$
3. $$\begin{array}{l}{\{(-3,3),(-2,1),(-1,-1),(0,-3)} , {(1,-5),(2,-4),(3,-2) \}}\end{array}$$

2. Function; not one-to-one

##### Exercise $$\PageIndex{4}$$ Determine Whether a Function is One-to-One

In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.

1. Figure 10.E.1

2. Figure 10.E.2

1. Figure 10.E.3

2. Figure 10.E.4

1.

1. Function; not one-to-one
2. Not a function
##### Exercise $$\PageIndex{5}$$ Find the Inverse of a Function

In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

1. $$\{(-3,10),(-2,5),(-1,2),(0,1)\}$$

1. Inverse function: $$\{(10,-3),(5,-2),(2,-1),(1,0)\}$$. Domain: $$\{1,2,5,10\}$$. Range: $$\{-3,-2,-1,0\}$$.

##### Exercise $$\PageIndex{6}$$ Find the Inverse of a Function

In the following exercise, graph the inverse of the one-to-one function shown.

##### Exercise $$\PageIndex{7}$$ Find the Inverse of a Function

In the following exercises, verify that the functions are inverse functions.

1. $$\begin{array}{l}{f(x)=3 x+7 \text { and }} {g(x)=\frac{x-7}{3}}\end{array}$$
2. $$\begin{array}{l}{f(x)=2 x+9 \text { and }} {g(x)=\frac{x+9}{2}}\end{array}$$

1. $$g(f(x))=x,$$ and $$f(g(x))=x,$$ so they are inverses.

##### Exercise $$\PageIndex{8}$$ Find the Inverse of a Function
1. $$f(x)=6 x-11$$
2. $$f(x)=x^{3}+13$$
3. $$f(x)=\frac{1}{x+5}$$
4. $$f(x)=\sqrt{x-1}$$

1. $$f^{-1}(x)=\frac{x+11}{6}$$

3. $$f^{-1}(x)=\frac{1}{x}-5$$

### Evaluate and Graph Exponential Functions

##### Exercise $$\PageIndex{9}$$ Graph Exponential Functions

In the following exercises, graph each of the following functions.

1. $$f(x)=4^{x}$$
2. $$f(x)=\left(\frac{1}{5}\right)^{x}$$
3. $$g(x)=(0.75)^{x}$$
4. $$g(x)=3^{x+2}$$
5. $$f(x)=(2.3)^{x}-3$$
6. $$f(x)=e^{x}+5$$
7. $$f(x)=-e^{x}$$

1.

3.

5.

7.

##### Exercise $$\PageIndex{10}$$ Solve Exponential Equations

In the following exercises, solve each equation.

1. $$3^{5 x-6}=81$$
2. $$2^{x^{2}}=16$$
3. $$9^{x}=27$$
4. $$5^{x^{2}+2 x}=\frac{1}{5}$$
5. $$e^{4 x} \cdot e^{7}=e^{19}$$
6. $$\frac{e^{x^{2}}}{e^{15}}=e^{2 x}$$

2. $$x=-2, x=2$$

4. $$x=-1$$

6. $$x=-3, x=5$$

##### Exercise $$\PageIndex{11}$$ Use Exponential Models in Applications

In the following exercises, solve.

1. Felix invested $$$12,000$$ in a savings account. If the interest rate is $$4$$% how much will be in the account in $$12$$ years by each method of compounding? 1. compound quarterly 2. compound monthly 3. compound continuously 2. Sayed deposits$$$20,000$$ in an investment account. What will be the value of his investment in $$30$$ years if the investment is earning $$7$$% per year and is compounded continuously?
3. A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. She starts her experiment with $$150$$ of the bacteria that grows at a rate of $$15$$% per hour. She will check on the bacteria every $$24$$ hours. How many bacteria will he find in $$24$$ hours?
4. In the last five years the population of the United States has grown at a rate of $$0.7$$% per year to about $$318,900,000$$. If this rate continues, what will be the population in $$5$$ more years?

2. $$\ 163,323.40$$

4. $$330,259,000$$

### Evaluate and Graph Logarithmic Functions

##### Exercise $$\PageIndex{12}$$ Convert Between Exponential and Logarithmic Form

In the following exercises, convert from exponential to logarithmic form.

1. $$5^{4}=625$$
2. $$10^{-3}=\frac{1}{1,000}$$
3. $$63^{\frac{1}{5}}=\sqrt{63}$$
4. $$e^{y}=16$$

2. $$\log \frac{1}{1,000}=-3$$

4. $$\ln 16=y$$

##### Exercise $$\PageIndex{13}$$ Convert Between Exponential and Logarithmic Form

In the following exercises, convert each logarithmic equation to exponential form.

1. $$7=\log _{2} 128$$
2. $$5=\log 100,000$$
3. $$4=\ln x$$

2. $$100000=10^{5}$$

##### Exercise $$\PageIndex{14}$$ Evaluate Logarithmic Functions

In the following exercises, solve for $$x$$.

1. $$\log _{x} 125=3$$
2. $$\log _{7} x=-2$$
3. $$\log _{\frac{1}{2}} \frac{1}{16}=x$$

1. $$x=5$$

3. $$x=4$$

##### Exercise $$\PageIndex{15}$$ Evaluate Logarithmic Functions

In the following exercises, find the exact value of each logarithm without using a calculator.

1. $$\log _{2} 32$$
2. $$\log _{8} 1$$
3. $$\log _{3} \frac{1}{9}$$

2. $$0$$

##### Exercise $$\PageIndex{16}$$ Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

1. $$y=\log _{5} x$$
2. $$y=\log _{\frac{1}{4}} x$$
3. $$y=\log _{0.8} x$$

1.

3.

##### Exercise $$\PageIndex{17}$$ Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

1. $$\log _{a} 36=5$$
2. $$\ln x=-3$$
3. $$\log _{2}(5 x-7)=3$$
4. $$\ln e^{3 x}=24$$
5. $$\log \left(x^{2}-21\right)=2$$

2. $$x=e^{-3}$$

4. $$x=8$$

##### Exercise $$\PageIndex{18}$$ Use Logarithmic Models in Applications

What is the decibel level of a train whistle with intensity $$10^{−3}$$ watts per square inch?

$$90$$ dB

### Use the Properties of Logarithms

##### Exercise $$\PageIndex{19}$$ Use the Properties of Logarithms

In the following exercises, use the properties of logarithms to evaluate.

1. $$\log _{7} 1$$
2. $$\log _{12} 12$$
1. $$5^{\log _{5} 13}$$
2. $$\log _{3} 3^{-9}$$
1. $$10^{\log \sqrt{5}}$$
2. $$\log 10^{-3}$$
1. $$e^{\ln 8}$$
2. $$\ln e^{5}$$

2.

1. $$13$$
2. $$-9$$

4.

1. $$8$$
2. $$5$$
##### Exercise $$\PageIndex{20}$$ Use the Properties of Logarithms

In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

1. $$\log _{4}(64 x y)$$
2. $$\log 10,000 m$$

2. $$4+\log m$$

##### Exercise $$\PageIndex{21}$$ Use the Properties of Logarithms

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

1. $$\log _{7} \frac{49}{y}$$
2. $$\ln \frac{e^{5}}{2}$$

2. $$5-\ln 2$$

##### Exercise $$\PageIndex{22}$$ Use the Properties of Logarithms

In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.

1. $$\log x^{-9}$$
2. $$\log _{4} \sqrt{z}$$

2. $$\frac{1}{7} \log _{4} z$$

##### Exercise $$\PageIndex{23}$$ Use the Properties of Logarithms

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

1. $$\log _{3}\left(\sqrt{4} x^{7} y^{8}\right)$$
2. $$\log _{5} \frac{8 a^{2} b^{6} c}{d^{3}}$$
3. $$\ln \frac{\sqrt{3 x^{2}-y^{2}}}{z^{4}}$$
4. $$\log _{6} \sqrt{\frac{7 x^{2}}{6 y^{3} z^{5}}}$$

2. $$\begin{array}{l}{\log _{5} 8+2 \log _{5} a+6 \log _{5} b} {+\log _{5} c-3 \log _{5} d}\end{array}$$

4. $$\begin{array}{l}{\frac{1}{3}\left(\log _{6} 7+2 \log _{6} x-1-3 \log _{6} y\right.} {-5 \log _{6} z )}\end{array}$$

##### Exercise $$\PageIndex{24}$$ Use the Properties of Logarithms

In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

1. $$\log _{2} 56-\log _{2} 7$$
2. $$3 \log _{3} x+7 \log _{3} y$$
3. $$\log _{5}\left(x^{2}-16\right)-2 \log _{5}(x+4)$$
4. $$\frac{1}{4} \log y-2 \log (y-3)$$

2. $$\log _{3} x^{3} y^{7}$$

4. $$\log \frac{\sqrt{y}}{(y-3)^{2}}$$

##### Exercise $$\PageIndex{25}$$ Use the Change-of-Base Formula

In the following exercises, rounding to three decimal places, approximate each logarithm.

1. $$\log _{5} 97$$
2. $$\log _{\sqrt{3}} 16$$

2. $$5.047$$

### Solve Exponential and Logarithmic Equations

##### Exercise $$\PageIndex{26}$$ Solve Logarithmic Equations Using the Properties of Logarithms

In the following exercises, solve for $$x$$.

1. $$3 \log _{5} x=\log _{5} 216$$
2. $$\log _{2} x+\log _{2}(x-2)=3$$
3. $$\log (x-1)-\log (3 x+5)=-\log x$$
4. $$\log _{4}(x-2)+\log _{4}(x+5)=\log _{4} 8$$
5. $$\ln (3 x-2)=\ln (x+4)+\ln 2$$

2. $$x=4$$

4. $$x=3$$

##### Exercise $$\PageIndex{27}$$ Solve Exponential Equations Using Logarithms

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

1. $$2^{x}=101$$
2. $$e^{x}=23$$
3. $$\left(\frac{1}{3}\right)^{x}=7$$
4. $$7 e^{x+3}=28$$
5. $$e^{x-4}+8=23$$

1. $$x=\frac{\log 101}{\log 2} \approx 6.658$$

3. $$x=\frac{\log 7}{\log \frac{1}{3}} \approx-1.771$$

5. $$x=\ln 15+4 \approx 6.708$$

##### Exercise $$\PageIndex{28}$$ Use Exponential Models in Applications
1. Jerome invests $$$18,000$$ at age $$17$$. He hopes the investments will be worth$$$30,000$$ when he turns $$26$$. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?
2. Elise invests $$$4500$$ in an account that compounds interest monthly and earns $$6$$%.How long will it take for her money to double? 3. Researchers recorded that a certain bacteria population grew from $$100$$ to $$300$$ in $$8$$ hours. At this rate of growth, how many bacteria will there be in $$24$$ hours? 4. Mouse populations can double in $$8$$ months $$\left(A=2 A_{0}\right)$$. How long will it take for a mouse population to triple? 5. The half-life of radioactive iodine is $$60$$ days. How much of a $$50$$ mg sample will be left in $$40$$ days? Answer 2. $$11.6$$ years 4. $$12.7$$ months ## Practice Test ##### Exercise $$\PageIndex{29}$$ 1. For the functions, $$f(x)=6x+1$$ and $$g(x)=8x−3$$, find 1. $$(f \circ g)(x)$$ 2. $$(g \circ f)(x)$$ 3. $$(f \cdot g)(x)$$ 2. Determine if the following set of ordered pairs represents a function and if so, is the function one-to-one. $$\{(-2,2),(-1,-3),(0,1),(1,-2),(2,-3)\}$$ 3. Determine whether each graph is the graph of a function and if so, is it one-to-one. 1. Figure 10.E.12 2. Figure 10.E.13 4. Graph, on the same coordinate system, the inverse of the one-to-one function shown. Figure 10.E.14 5. Find the inverse of the function $$f(x)=x^{5}−9$$. 6. Graph the function $$g(x)=2^{x-3}$$. 7. Solve the equation $$2^{2 x-4}=64$$. 8. Solve the equation $$\frac{e^{x^{2}}}{e^{4}}=e^{3 x}$$. 9. Megan invested$$$21,000$$ in a savings account. If the interest rate is $$5$$%, how much will be in the account in $$8$$ years by each method of compounding?

1. compound quarterly
2. compound monthly
3. compound continuously

10. Convert the equation from exponential to logarithmic form: $$10^{-2}=\frac{1}{100}$$.

11. Convert the equation from logarithmic equation to exponential form: $$3=\log _{7} 343$$.

12. Solve for $$x$$: $$\log _{5} x=-3$$

13. Evaluate log $$_{11} 1$$.

14. Evaluate $$\log _{4} \frac{1}{64}$$.

15. Graph the function $$y=\log _{3} x$$.

16. Solve for $$x$$: $$\log \left(x^{2}-39\right)=1$$

17. What is the decibel level of a small fan with intensity $$10^{−8}$$ watts per square inch?

18. Evaluate each.

1. $$6^{\log _{6} 17}$$
2. $$\log _{9} 9^{-3}$$

1.

1. $$48 x-17$$
2. $$48 x+5$$
3. $$48 x^{2}-10 x-3$$

3.

1. Not a function
2. One-to-one function

5. $$f^{-1}(x)=\sqrt{x+9}$$

7. $$x=5$$

9.

1. $$$31,250.74$$ 2.$$$31,302.29$$
3. $$$31,328.32$$ 11. $$343=7^{3}$$ 13. $$0$$ 15. Figure 10.E.15 17. $$40$$ dB ##### Exercise $$\PageIndex{30}$$ In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible. 1. $$\log _{5} 25 a b$$ 2. $$\ln \frac{e^{12}}{8}$$ 3. $$\log _{2} \sqrt{\frac{5 x^{3}}{16 y^{2} z^{7}}}$$ Answer 1. $$2+\log _{5} a+\log _{5} b$$ 3. $$\begin{array}{l}{\frac{1}{4}\left(\log _{2} 5+3 \log _{2} x-4-2 \log _{2} y\right.} {-7 \log _{2} z )}\end{array}$$ ##### Exercise $$\PageIndex{31}$$ In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible. 1. $$5 \log _{4} x+3 \log _{4} y$$ 2. $$\frac{1}{6} \log x-3 \log (x+5)$$ 3. Rounding to three decimal places, approximate $$\log _{4} 73$$. 4. Solve for $$x$$: $$\log _{7}(x+2)+\log _{7}(x-3)=\log _{7} 24$$ Answer 2. $$\log \frac{\sqrt{x}}{(x+5)^{3}}$$ 4. $$x=6$$ ##### Exercise $$\PageIndex{32}$$ In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. 1. $$\left(\frac{1}{5}\right)^{x}=9$$ 2. $$5 e^{x-4}=40$$ 3. Jacob invests$$$14,000$$ in an account that compounds interest quarterly and earns $$4$$%. How long will it take for his money to double?
4. Researchers recorded that a certain bacteria population grew from $$500$$ to $$700$$ in $$5$$ hours. At this rate of growth, how many bacteria will there be in $$20$$ hours?
5. A certain beetle population can double in $$3$$ months $$\left(A=2 A_{0}\right)$$. How long will it take for that beetle population to triple?
2. $$x=\ln 8+4 \approx 6.079$$
4. $$1,921$$ bacteria