$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 10.4E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Section 10.3 Exercises

### Practice Makes Perfect

##### Exercise $$\PageIndex{21}$$ Convert Between Exponential and Logarithmic Form

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

1. $$4^{2}=16$$
2. $$2^{5}=32$$
3. $$3^{3}=27$$
4. $$5^{3}=125$$
5. $$10^{3}=1000$$
6. $$10^{-2}=\frac{1}{100}$$
7. $$x^{\frac{1}{2}}=\sqrt{3}$$
8. $$x^{\frac{1}{3}}=\sqrt[3]{6}$$
9. $$32^{x}=\sqrt[4]{32}$$
10. $$17^{x}=\sqrt[5]{17}$$
11. $$\left(\frac{1}{4}\right)^{2}=\frac{1}{16}$$
12. $$\left(\frac{1}{3}\right)^{4}=\frac{1}{81}$$
13. $$3^{-2}=\frac{1}{9}$$
14. $$4^{-3}=\frac{1}{64}$$
15. $$e^{x}=6$$
16. $$e^{3}=x$$

2. $$\log _{2} 32=5$$

4. $$\log _{5} 125=3$$

6. $$\log \frac{1}{100}=-2$$

8. $$\log _{x} \sqrt[3]{6}=\frac{1}{3}$$

10. $$\log _{17} \sqrt[5]{17}=x$$

12. $$\log _{\frac{1}{3}} \frac{1}{81}=4$$

14. $$\log _{4} \frac{1}{64}=-3$$

16. $$\ln x=3$$

##### Exercise $$\PageIndex{22}$$ Convert Between Exponential and Logarithmic Form

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

1. $$3=\log _{4} 64$$
2. $$6=\log _{2} 64$$
3. $$4=\log _{x} 81$$
4. $$5=\log _{x} 32$$
5. $$0=\log _{12} 1$$
6. $$0=\log _{7} 1$$
7. $$1=\log _{3} 3$$
8. $$1=\log _{9} 9$$
9. $$-4=\log _{10} \frac{1}{10,000}$$
10. $$3=\log _{10} 1,000$$
11. $$5=\log _{e} x$$
12. $$x=\log _{e} 43$$

2. $$64=2^{6}$$

4. $$32=x^{5}$$

6. $$1=7^{0}$$

8. $$9=9^{1}$$

10. $$1,000=10^{3}$$

12. $$43=e^{x}$$

##### Exercise $$\PageIndex{23}$$ Evaluate Logarithmic Functions

In the following exercises, find the value of $$x$$ in each logarithmic equation.

1. $$\log _{x} 49=2$$
2. $$\log _{x} 121=2$$
3. $$\log _{x} 27=3$$
4. $$\log _{x} 64=3$$
5. $$\log _{3} x=4$$
6. $$\log _{5} x=3$$
7. $$\log _{2} x=-6$$
8. $$\log _{3} x=-5$$
9. $$\log _{\frac{1}{4}} \frac{1}{16}=x$$
10. $$\log _{\frac{1}{3}} \frac{1}{9}=x$$
11. $$\log _{\frac{1}{4}} 64=x$$
12. $$\log _{\frac{1}{9}} 81=x$$

2. $$x=11$$

4. $$x=4$$

6. $$x=125$$

8. $$x=\frac{1}{243}$$

10. $$x=2$$

12. $$x=-2$$

##### Exercise $$\PageIndex{24}$$ Evaluate Logarithmic Functions

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

1. $$\log _{7} 49$$
2. $$\log _{6} 36$$
3. $$\log _{4} 1$$
4. $$\log _{5} 1$$
5. $$\log _{16} 4$$
6. $$\log _{27} 3$$
7. $$\log _{\frac{1}{2}} 2$$
8. $$\log _{\frac{1}{2}} 4$$
9. $$\log _{2} \frac{1}{16}$$
10. $$\log _{3} \frac{1}{27}$$
11. $$\log _{4} \frac{1}{16}$$
12. $$\log _{9} \frac{1}{81}$$

2. $$2$$

4. $$0$$

6. $$\frac{1}{3}$$

8. $$-2$$

10. $$-3$$

12. $$-2$$

##### Exercise $$\PageIndex{25}$$ Graph Logarithmic Functions

In the following exercises, graph each logarithmic function.

1. $$y=\log _{2} x$$
2. $$y=\log _{4} x$$
3. $$y=\log _{6} x$$
4. $$y=\log _{7} x$$
5. $$y=\log _{1.5} x$$
6. $$y=\log _{2.5} x$$
7. $$y=\log _{\frac{1}{3}} x$$
8. $$y=\log _{\frac{1}{5}} x$$
9. $$y=\log _{0.4} x$$
10. $$y=\log _{0.6} x$$

2.

4.

6.

8.

10.

##### Exercise $$\PageIndex{26}$$ Solve Logarithmic Equations

In the following exercises, solve each logarithmic equation.

1. $$\log _{a} 16=2$$
2. $$\log _{a} 81=2$$
3. $$\log _{a} 8=3$$
4. $$\log _{a} 27=3$$
5. $$\log _{a} 32=2$$
6. $$\log _{a} 24=3$$
7. $$\ln x=5$$
8. $$\ln x=4$$
9. $$\log _{2}(5 x+1)=4$$
10. $$\log _{2}(6 x+2)=5$$
11. $$\log _{3}(4 x-3)=2$$
12. $$\log _{3}(5 x-4)=4$$
13. $$\log _{4}(5 x+6)=3$$
14. $$\log _{4}(3 x-2)=2$$
15. $$\ln e^{4 x}=8$$
16. $$\ln e^{2 x}=6$$
17. $$\log x^{2}=2$$
18. $$\log \left(x^{2}-25\right)=2$$
19. $$\log _{2}\left(x^{2}-4\right)=5$$
20. $$\log _{3}\left(x^{2}+2\right)=3$$

2. $$a=9$$

4. $$a=3$$

6. $$a=\sqrt[3]{24}$$

8. $$x=e^{4}$$

10. $$x=5$$

12. $$x=17$$

14. $$x=6$$

16. $$x=3$$

18. $$x=-5 \sqrt{5}, x=5 \sqrt{5}$$

20. $$x=-5, x=5$$

##### Exercise $$\PageIndex{27}$$ Use Logarithmic Models in Applications

In the following exercises, use a logarithmic model to solve.

1. What is the decibel level of normal conversation with intensity $$10^{−6}$$ watts per square inch?
2. What is the decibel level of a whisper with intensity $$10^{−10}$$ watts per square inch?
3. What is the decibel level of the noise from a motorcycle with intensity $$10^{−2}$$ watts per square inch?
4. What is the decibel level of the sound of a garbage disposal with intensity $$10^{−2}$$ watts per square inch?
5. In 2014, Chile experienced an intense earthquake with a magnitude of $$8.2$$ on the Richter scale. In 2010, Haiti also experienced an intense earthquake which measured $$7.0$$ on the Richter scale. Compare the intensities of the two earthquakes.
6. The Los Angeles area experiences many earthquakes. In 1994, the Northridge earthquake measured magnitude of $$6.7$$ on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured $$5.1$$ on the Richter scale. Compare the intensities of the two earthquakes.

2. A whisper has a decibel level of $$20$$ dB.

4. The sound of a garbage disposal has a decibel level of $$100$$ dB.

6. The intensity of the 1994 Northridge earthquake in the Los Angeles area was about $$40$$ times the intensity of the 2014 earthquake.

##### Exercise $$\PageIndex{28}$$ Writing Exercises
1. Explain how to change an equation from logarithmic form to exponential form.
2. Explain the difference between common logarithms and natural logarithms.
3. Explain why $$\log _{a} a^{x}=x$$.
4. Explain how to find the $$\log _{7} 32$$ on your calculator.