# 10.6E: Exercises

• • OpenStax
• OpenStax
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

### Practice Makes Perfect

##### Exercise $$\PageIndex{17}$$ Solve Logarithmic Equations Using the Properties of Logarithms

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

1. $$\log _{4} 64=2 \log _{4} x$$
2. $$\log 49=2 \log x$$
3. $$3 \log _{3} x=\log _{3} 27$$
4. $$3 \log _{6} x=\log _{6} 64$$
5. $$\log _{5}(4 x-2)=\log _{5} 10$$
6. $$\log _{3}\left(x^{2}+3\right)=\log _{3} 4 x$$
7. $$\log _{3} x+\log _{3} x=2$$
8. $$\log _{4} x+\log _{4} x=3$$
9. $$\log _{2} x+\log _{2}(x-3)=2$$
10. $$\log _{3} x+\log _{3}(x+6)=3$$
11. $$\log x+\log (x+3)=1$$
12. $$\log x+\log (x-15)=2$$
13. $$\log (x+4)-\log (5 x+12)=-\log x$$
14. $$\log (x-1)-\log (x+3)=\log \frac{1}{x}$$
15. $$\log _{5}(x+3)+\log _{5}(x-6)=\log _{5} 10$$
16. $$\log _{5}(x+1)+\log _{5}(x-5)=\log _{5} 7$$
17. $$\log _{3}(2 x-1)=\log _{3}(x+3)+\log _{3} 3$$
18. $$\log (5 x+1)=\log (x+3)+\log 2$$

2. $$x=7$$

4. $$x=4$$

6. $$x=1, x=3$$

8. $$x=8$$

10. $$x=3$$

12. $$x=20$$

14. $$x=3$$

16. $$x=6$$

18. $$x=\frac{5}{3}$$

##### Exercise $$\PageIndex{18}$$ 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. $$3^{x}=89$$
2. $$2^{x}=74$$
3. $$5^{x}=110$$
4. $$4^{x}=112$$
5. $$e^{x}=16$$
6. $$e^{x}=8$$
7. $$\left(\frac{1}{2}\right)^{x}=6$$
8. $$\left(\frac{1}{3}\right)^{x}=8$$
9. $$4 e^{x+1}=16$$
10. $$3 e^{x+2}=9$$
11. $$6 e^{2 x}=24$$
12. $$2 e^{3 x}=32$$
13. $$\frac{1}{4} e^{x}=3$$
14. $$\frac{1}{3} e^{x}=2$$
15. $$e^{x+1}+2=16$$
16. $$e^{x-1}+4=12$$

2. $$x=\frac{\log 74}{\log 2} \approx 6.209$$

4. $$x=\frac{\log 112}{\log 4} \approx 3.404$$

6. $$x=\ln 8 \approx 2.079$$

8. $$x=\frac{\log 8}{\log \frac{1}{3}} \approx-1.893$$

10. $$x=\ln 3-2 \approx-0.901$$

12. $$x=\frac{\ln 16}{3} \approx 0.924$$

14. $$x=\ln 6 \approx 1.792$$

16. $$x=\ln 8+1 \approx 3.079$$

##### Exercise $$\PageIndex{19}$$ Solve Exponential Equations Using Logarithms

In the following exercises, solve each equation.

1. $$3^{3 x+1}=81$$
2. $$6^{4 x-17}=216$$
3. $$\frac{e^{x^{2}}}{e^{14}}=e^{5 x}$$
4. $$\frac{e^{x^{2}}}{e^{x}}=e^{20}$$
5. $$\log _{a} 64=2$$
6. $$\log _{a} 81=4$$
7. $$\ln x=-8$$
8. $$\ln x=9$$
9. $$\log _{5}(3 x-8)=2$$
10. $$\log _{4}(7 x+15)=3$$
11. $$\ln e^{5 x}=30$$
12. $$\ln e^{6 x}=18$$
13. $$3 \log x=\log 125$$
14. $$7 \log _{3} x=\log _{3} 128$$
15. $$\log _{6} x+\log _{6}(x-5)=\log _{6} 24$$
16. $$\log _{9} x+\log _{9}(x-4)=\log _{9} 12$$
17. $$\log _{2}(x+2)-\log _{2}(2 x+9)=-\log _{2} x$$
18. $$\log _{6}(x+1)-\log _{6}(4 x+10)=\log _{6} \frac{1}{x}$$

2. $$x=5$$

4. $$x=-4, x=5$$

6. $$a=3$$

8. $$x=e^{9}$$

10. $$x=7$$

12. $$x=3$$

14. $$x=2$$

16. $$x=6$$

18. $$x=5$$

##### Exercise $$\PageIndex{20}$$ Solve Exponential Equations Using Logarithms

In the following exercises, solve for $$x$$, giving an exact answer as well as an approximation to three decimal places.

1. $$6^{x}=91$$
2. $$\left(\frac{1}{2}\right)^{x}=10$$
3. $$7 e^{x-3}=35$$
4. $$8 e^{x+5}=56$$

2. $$x=\frac{\log 10}{\log \frac{1}{2}} \approx-3.322$$

4. $$x=\ln 7-5 \approx-3.054$$

##### Exercise $$\PageIndex{21}$$ Use Exponential Models in Applications

In the following exercises, solve.

1. Sung Lee invests $$$5,000$$ at age $$18$$. He hopes the investments will be worth$$$10,000$$ when he turns $$25$$. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?
2. Alice invests $$$15,000$$ at age $$30$$ from the signing bonus of her new job. She hopes the investments will be worth$$$30,000$$ when she turns $$40$$. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?
3. Coralee invests $$$5,000$$ in an account that compounds interest monthly and earns $$7$$%. How long will it take for her money to double? 4. Simone invests$$$8,000$$ in an account that compounds interest quarterly and earns $$5$$%. How long will it take for his money to double?
5. Researchers recorded that a certain bacteria population declined from $$100,000$$ to $$100$$ in $$24$$ hours. At this rate of decay, how many bacteria will there be in $$16$$ hours?
6. Researchers recorded that a certain bacteria population declined from $$800,000$$ to $$500,000$$ in $$6$$ hours after the administration of medication. At this rate of decay, how many bacteria will there be in $$24$$ hours?
7. A virus takes $$6$$ days to double its original population $$\left(A=2 A_{0}\right)$$. How long will it take to triple its population?
8. A bacteria doubles its original population in $$24$$ hours $$\left(A=2 A_{0}\right)$$. How big will its population be in $$72$$ hours?
9. Carbon-14 is used for archeological carbon dating. Its half-life is $$5,730$$ years. How much of a $$100$$-gram sample of Carbon-14 will be left in $$1000$$ years?
10. Radioactive technetium-99m is often used in diagnostic medicine as it has a relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is $$6$$ hours, how much of the radioactive material form a $$0.5$$ ml injection will be in the body in $$24$$ hours?

2. $$6.9$$%

4. $$13.9$$ years

6. $$122,070$$ bacteria

8. $$8$$ times as large as the original population

10. $$0.03$$ mL

##### Exercise $$\PageIndex{22}$$ Writing Exercises
1. Explain the method you would use to solve these equations: $$3^{x+1}=81$$, $$3^{x+1}=75$$. Does your method require logarithms for both equations? Why or why not?
2. What is the difference between the equation for exponential growth versus the equation for exponential decay?