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# 7.E: Exponential and Logarithmic Functions (Exercises)

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Exercise $$\PageIndex{1}$$

Given $$f$$ and $$g$$ find $$(f \circ g)(x)$$ and $$(g \circ f)(x)$$.

1. $$f(x)=6 x-5, g(x)=2 x+1$$
2. $$f(x)=5-6 x, g(x)=\frac{3}{2} x$$
3. $$f(x)=2 x^{2}+x-2, g(x)=5 x$$
4. $$f(x)=x^{2}-x-6, g(x)=x-3$$
5. $$f(x)=\sqrt{x+2}, g(x)=8 x-2$$
6. $$f(x)=\frac{x-1}{3 x-1}, g(x)=\frac{1}{x}$$
7. $$f(x)=x^{2}+3 x-1, g(x)=\frac{1}{x-2}$$
8. $$f(x)=\sqrt{3(x+2)}, g(x)=9 x^{3}-2$$

1. $$(f \circ g)(x)=12 x+1 ;(g \circ f)(x)=12 x-9$$

3. $$\begin{array}{l}{(f \circ g)(x)=50 x^{2}+5 x-2}; \: {(g \circ f)(x)=10 x^{2}+5 x-10}\end{array}$$

5. $$(f\circ g)(x)=2\sqrt{2x};\:(g\circ f)(x)=8\sqrt{x+2}-2$$

7. $$\begin{array}{c}{(f \circ g)(x)=-\frac{x^{2}-7 x+9}{(x-2)^{2}}}; \: {\left(g \circ f\right)(x)=\frac{1}{x^{2}+3 x-3}}\end{array}$$

Exercise $$\PageIndex{1}$$

Are the given functions one-to-one? Explain.

1.

2.

3.

4.

1. No, fails the HLT

3. Yes, passes the HLT

Exercise $$\PageIndex{3}$$

Verify algebraically that the two given functions are inverses. In other words, show that $$\left(f \circ f^{-1}\right)(x)=x$$ and $$\left(f^{-1} \circ f\right)(x)=x$$.

1. $$f(x)=6 x-5, f^{-1}(x)=\frac{1}{6} x+\frac{5}{6}$$
2. $$f(x)=\sqrt{2 x+3}, f^{-1}(x)=\frac{x^{2}-3}{2}, x \geq 0$$
3. $$f(x)=\frac{x}{3 x-2}, f^{-1}(x)=\frac{2 x}{3 x-1}$$
4. $$f(x)=\sqrt{x+3}-4, f^{-1}(x)=(x+4)^{3}-3$$

1. Proof

3. Proof

Exercise $$\PageIndex{4}$$

Find the inverses of each function defined as follows:

1. $$f(x)=-7 x+3$$
2. $$f(x)=\frac{2}{3} x-\frac{1}{2}$$
3. $$g(x)=x^{2}-12, x \geq 0$$
4. $$g(x)=(x-1)^{3}+5$$
5. $$g(x)=\frac{2}{x-1}$$
6. $$h(x)=\frac{x+5}{x-5}$$
7. $$h(x)=\frac{3 x-1}{x}$$
8. $$p(x)=\sqrt{5 x}+3$$
9. $$h(x)=\sqrt{2 x-7}+2$$
10. $$h(x)=\sqrt{x+2}-3$$

1. $$f^{-1}(x)=-\frac{1}{7} x+\frac{3}{7}$$

3. $$g^{-1}(x)=\sqrt{x+12}$$

5. $$g^{-1}(x)=\frac{x+2}{x}$$

7. $$h^{-1}(x)=-\frac{1}{x-3}$$

9. $$h^{-1}(x)=\frac{(x-2)^{3}+7}{2}$$

Exercise $$\PageIndex{5}$$

Evaluate.

1. $$f(x)=5^{x} ;$$ find $$f(-1), f(0),$$ and $$f(3).$$
2. $$f(x)=\left(\frac{1}{2}\right)^{x} ;$$ find $$f(-4), f(0),$$ and $$f(-3).$$
3. $$g(x)=10^{-x} ;$$ find $$g(-5), g(0),$$ and $$g(2).$$
4. $$g(x)=1-3^{x} ;$$ find $$g(-2), g(0),$$ and $$g(3).$$

1. $$f(-1)=\frac{1}{5}, f(0)=1, f(3)=125$$

3. $$g(-5)=100,000, g(0)=1, g(2)=\frac{1}{100}$$

Exercise $$\PageIndex{6}$$

Sketch the exponential function. Draw the horizontal asymptote with a dashed line.

1. $$f(x)=5^{x}+10$$
2. $$f(x)=5^{x-4}$$
3. $$f(x)=-3^{x}-9$$
4. $$f(x)=3^{x+2}+6$$
5. $$f(x)=\left(\frac{1}{3}\right)^{x}$$
6. $$f(x)=\left(\frac{1}{2}\right)^{x}-4$$
7. $$f(x)=2^{-x}+3$$
8. $$f(x)=1-3^{-x}$$

1.

3.

5.

7.

Exercise $$\PageIndex{7}$$

Use a calculator to evaluate the following. Round off to the nearest hundredth.

1. $$f(x)=e^{x}+1 ;$$ find $$f(-3), f(-1),$$ and $$f\left(\frac{1}{2}\right)$$.
2. $$g(x)=2-3 e^{x} ;$$ find $$g(-1), g(0),$$ and $$g\left(\frac{2}{3}\right)$$.
3. $$p(x)=1-5 e^{-x} ;$$ find $$p(-4), p\left(-\frac{1}{2}\right),$$ and $$p(0)$$.
4. $$r(x)=e^{-2 x}-1 ;$$ find $$r(-1), r\left(\frac{1}{4}\right),$$ and $$r(2)$$.

1. $$f(-3) \approx 1.05, f(-1) \approx 1.37, f\left(\frac{1}{2}\right) \approx 2.65$$

3. $$p(-4) \approx-271.99, p\left(-\frac{1}{2}\right) \approx-7.24, p(0)=-4$$

Exercise $$\PageIndex{8}$$

Sketch the function. Draw the horizontal asymptote with a dashed line.

1. $$f(x)=e^{x}+4$$
2. $$f(x)=e^{x-4}$$
3. $$f(x)=e^{x+3}+2$$
4. $$f(x)=e^{-x}+5$$
5. Jerry invested $$$6,250$$ in an account earning $$3 \frac{5}{8}$$% annual interest that is compounded monthly. How much will be in the account after $$4$$ years? 6. Jose invested$$$7,500$$ in an account earning $$4 \frac{1}{4}$$% annual interest that is compounded continuously. How much will be in the account after $$3 \frac{1}{2}$$ years?
7. A $$14$$-gram sample of radioactive iodine is accidently released into the atmosphere. The amount of the substance in grams is given by the formula $$P (t) = 14e^{ −0.087t}$$, where $$t$$ represents the time in days after the sample was released. How much radioactive iodine will be present in the atmosphere $$30$$ days after it was released?
8. The number of cells in a bacteria sample is given by the formula $$N(t)=\frac{2.4 \times 10^{5}}{1+9 e^{-0.28t}}$$, where $$t$$ represents the time in hours since the initial placement of $$24,000$$ cells. Use the formula to calculate the number of cells in the sample $$20$$ hours later.

1.

3.

5. $$$7,223.67$$ 7. Approximately $$1$$ gram Exercise $$\PageIndex{9}$$ Evaluate. 1. $$\log _{4} 16$$ 2. $$\log _{3} 27$$ 3. $$\log _{2}\left(\frac{1}{32}\right)$$ 4. $$\log \left(\frac{1}{10}\right)$$ 5. $$\log _{1 / 3} 9$$ 6. $$\log _{3 / 4}\left(\frac{4}{3}\right)$$ 7. $$\log _{7} 1$$ 8. $$\log _{3}(-3)$$ 9. $$\log _{4} 0$$ 10. $$\log _{3} 81$$ 11. $$\log _{6} \sqrt{6}$$ 12. $$\log _{5} \sqrt{25}$$ 13. $$\ln e^{8}$$ 14. $$\ln \left(\frac{1}{e^{5}}\right)$$ 15. $$\log (0.00001)$$ 16. $$\log 1,000,000$$ Answer 1. $$2$$ 3. $$−5$$ 5. $$−2$$ 7. $$0$$ 9. Undefined 11. $$\frac{1}{2}$$ 13. $$8$$ 15. $$−5$$ Exercise $$\PageIndex{10}$$ Find $$x$$. 1. $$\log _{5} x=3$$ 2. $$\log _{3} x=-4$$ 3. $$\log _{2 / 3} x=3$$ 4. $$\log _{3} x=\frac{2}{5}$$ 5. $$\log x=-3$$ 6. $$\ln x=\frac{1}{2}$$ Answer 1. $$125$$ 3. $$\frac{8}{27}$$ 5. $$0.001$$ Exercise $$\PageIndex{11}$$ Sketch the graph of the logarithmic function. Draw the vertical asymptote with a dashed line. 1. $$f(x)=\log _{2}(x-5)$$ 2. $$f(x)=\log _{2} x-5$$ 3. $$g(x)=\log _{3}(x+5)+15$$ 4. $$g(x)=\log _{3}(x-5)-5$$ 5. $$h(x)=\log _{4}(-x)+1$$ 6. $$h(x)=3-\log _{4} x$$ 7. $$g(x)=\ln (x-2)+3$$ 8. $$g(x)=\ln (x+3)-1$$ 9. The population of a certain small town is growing according to the function $$P (t) = 89,000(1.035)^{t}$$, where $$t$$ represents time in years since the last census. Use the function to estimate the population $$8 \frac{1}{2}$$ years after the census was taken. 10. The volume of sound $$L$$ in decibels (dB) is given by the formula $$L=10 \log \left(I / 10^{-12}\right)$$, where $$I$$ represents the intensity of the sound in watts per square meter. Determine the volume of a sound with an intensity of $$0.5$$ watts per square meter. Answer 1. Figure 7.E.11 3. Figure 7.E.12 5. Figure 7.E.13 7. Figure 7.E.14 9. $$119,229$$ people Exercise $$\PageIndex{12}$$ Evaluate without using a calculator. 1. $$\log _{9} 9$$ 2. $$\log _{8} 1$$ 3. $$\log _{1 / 3} 3$$ 4. $$\log \left(\frac{1}{10}\right)$$ 5. $$e^{\ln 17}$$ 6. $$10^{\log 27}$$ 7. $$\ln e^{63}$$ 8. $$\log 10^{33}$$ Answer 1. $$1$$ 3. $$−1$$ 5. $$17$$ 7. $$63$$ Exercise $$\PageIndex{13}$$ Expand completely. 1. $$\log \left(100 x^{2}\right)$$ 2. $$\log _{5}\left(5 x^{3}\right)$$ 3. $$\log _{3}\left(\frac{3 x^{5}}{5}\right)$$ 4. $$\ln \left(\frac{10}{3 x^{2}}\right)$$ 5. $$\log _{2}\left(\frac{8 x^{2}}{y^{2} z}\right)$$ 6. $$\log \left(\frac{x^{10}}{10 y^{3} z^{4}}\right)$$ 7. $$\ln \left(\frac{3 b \sqrt{a}}{c^{4}}\right)$$ 8. $$\log \left(\frac{20 y^{3}}{\sqrt{x^{2}}}\right)$$ Answer 1. $$2+2 \log x$$ 3. $$1+5 \log _{3} x-\log _{3} 5$$ 5. $$3+2 \log _{2} x-2 \log _{2} y-\log _{2} z$$ 7. $$\ln 3+\ln b+\frac{1}{2} \ln a-4 \ln c$$ Exercise $$\PageIndex{14}$$ Write as a single logarithm with coefficient $$1$$. 1. $$\log x+2 \log y-3 \log z$$ 2. $$\log _{2} 5-3 \log _{2} x+4 \log _{2} y$$ 3. $$-2 \log _{5} x+\log _{5} y-5 \log _{5}(x-1)$$ 4. $$\ln x-\ln (x-1)-\ln (x+1)$$ 5. $$3 \log _{2} x+\frac{1}{2} \log _{2} y-\frac{2}{3} \log _{2} z$$ 6. $$\frac{1}{3} \log x-3 \log y-\frac{3}{5} \log z$$ 7. $$\log _{5} 4+5 \log _{5} x-\frac{1}{3}\left(\log _{5} y+2 \log _{5} z\right)$$ 8. $$\ln x-\frac{1}{2}(\ln y-4 \ln z)$$ Answer 1. $$\log \left(\frac{x y^{2}}{z^{3}}\right)$$ 3. $$\log _{5}\left(\frac{y}{x^{2}(x-1)^{5}}\right)$$ 5. $$\log _{2}\left(\frac{x^{3} \sqrt{y}}{\sqrt{z^{2}}}\right)$$ 7. $$\log _{5}\left(\frac{4 x^{5}}{\sqrt{y z^{2}}}\right)$$ Exercise $$\PageIndex{15}$$ Solve. Give the exact answer and the approximate answer rounded to the nearest hundredth where appropriate. 1. $$5^{2 x+1}=125$$ 2. $$10^{3 x-2}=100$$ 3. $$9^{x-3}=81$$ 4. $$16^{2 x+3}=8$$ 5. $$5^{x}=7$$ 6. $$3^{2 x}=5$$ 7. $$10^{x+2}-3=7$$ 8. $$e^{2 x-1}+2=3$$ 9. $$7^{4 x-1}-2=9$$ 10. $$3^{5 x-2}+5=7$$ 11. $$3-e^{4 x}=2$$ 12. $$5+e^{3 x}=4$$ 13. $$\frac{4}{1+e^{5 x}}=2$$ 14. $$\frac{100}{1+e^{3 x}}=\frac{1}{2}$$ Answer 1. $$1$$ 3. $$5$$ 5. $$\frac{\log (7)}{\log (5)} \approx 1.21$$ 7. $$-1$$ 9. $$\frac{\log 7+\log 11}{4 \log 7} \approx 0.56$$ 11. $$0$$ 13. $$0$$ Exercise $$\PageIndex{16}$$ Use the change of base formula to approximate the following to the nearest tenth. 1. $$\log _{5} 13$$ 2. $$\log _{2} 27$$ 3. $$\log _{4} 5$$ 4. $$\log _{9} 0.81$$ 5. $$\log _{1 / 4} 21$$ 6. $$\log _{2} \sqrt{5}$$ Answer 1. $$1.6$$ 3. $$1.2$$ 5. $$-2.2$$ Exercise $$\PageIndex{17}$$ Solve. 1. $$\log _{2}(3 x-5)=\log _{2}(2 x+7)$$ 2. $$\ln (7 x)=\ln (x+8)$$ 3. $$\log _{5} 8-2 \log _{5} x=\log _{5} 2$$ 4. $$\log _{3}(x+2)+\log _{3}(x)=\log _{3} 8$$ 5. $$\log _{5}(2 x-1)=2$$ 6. $$2 \log _{4}(3 x-2)=4$$ 7. $$2=\log _{2}\left(x^{2}-4\right)-\log _{2} 3$$ 8. $$\log _{2}(x-1)+\log _{2}(x+1)=3$$ 9. $$\log _{2} x+\log _{2}(x-1)=1$$ 10. $$\log _{4}(x+5)+\log _{4}(x+11)=2$$ 11. $$\log (2 x+5)-\log (x-1)=1$$ 12. $$\ln x-\ln (2 x-1)=1$$ 13. $$2 \log _{2}(x+4)=\log _{2}(x+2)+3$$ 14. $$2 \log _{3} x=1+\log _{3}(x+6)$$ 15. $$\log _{3}(x+1)-2 \log _{3} x=1$$ 16. $$\log _{5}(2 x)+\log _{5}(x-1)=1$$ Answer 1. $$12$$ 3. $$2$$ 5. $$13$$ 7. $$±4$$ 9. $$2$$ 11. $$\frac{15}{8}$$ 13. $$0$$ 15. $$\frac{1+\sqrt{13}}{6}$$ Exercise $$\PageIndex{18}$$ Solve. 1. An amount of$$$3,250$$ is invested in an account that earns $$4.6$$% annual interest that is compounded monthly. Estimate the number of years for the amount in the account to reach $$$4,000$$. 2. An amount of$$$2,500$$ is invested in an account that earns $$5.5$$% annual interest that is compounded continuously. Estimate the number of years for the amount in the account to reach $$$3,000$$. 3. How long does it take to double an investment made in an account that earns $$6 \frac{3}{4}$$% annual interest that is compounded continuously? 4. How long does it take to double an investment made in an account that earns $$6 \frac{3}{4}$$% annual interest that is compounded semi-annually? 5. In the year 2000 a certain small town had a population of $$46,000$$ people. In the year 2010 the population was estimated to have grown to $$92,000$$ people. If the population continues to grow exponentially at this rate, estimate the population in the year 2016. 6. A fleet van was purchased new for$$$28,000$$ and $$2$$ years later it was valued at $$$20,000$$. If the value of the van continues to decrease exponentially at this rate, determine its value $$7$$ years after it is purchased new. 7. A website that has been in decline registered $$4,200$$ unique visitors last month and $$3,600$$ unique visitors this month. If the number of unique visitors continues to decline exponentially, how many unique visitors would you expect next month? 8. An initial population of $$18$$ rabbits was introduced into a wildlife preserve. The number of rabbits doubled in the first year. If the rabbit population continues to grow exponentially at this rate, how many rabbits will be present $$5$$ years after they were introduced? 9. The half-life of sodium-24 is about $$15$$ hours. How long will it take a $$50$$-milligram sample to decay to $$10$$ milligrams? 10. The half-life of radium-226 is about $$1,600$$ years. How long will it take an initial sample to decay to $$30$$% of the original amount? 11. An archeologist discovered a bone tool artifact. After analysis, the artifact was found to contain $$62$$% of the carbon-14 normally found in bone from the same animal. Given that carbon-14 has a half-life of $$5,730 years$$, estimate the age of the artifact. 12. The half-life of radioactive iodine-131 is about $$8$$ days. What percentage of an initial sample accidentally released into the atmosphere do we expect to remain after $$53$$ days? Answer 1. $$4.5$$ years 3. $$10.27$$ years 5. About $$139,446$$ people 7. $$3,086$$ unique visitors 9. $$35$$ hours 11. About $$3,952$$ years old ## Sample Exam Exercise $$\PageIndex{19}$$ 1. Given $$f(x)=x^{2}-x+3$$ and $$g(x)=3 x-1$$ find $$(f \circ g)(x)$$. 2. Show that $$f(x)=\sqrt{7 x-2}$$ and $$g(x)=\frac{x^{3}+2}{7}$$ are inverses. Answer 1. $$(f \circ g)(x)=9 x^{2}-9 x+5$$ Exercise $$\PageIndex{20}$$ Find the inverse of the following functions: 1. $$f(x)=\frac{1}{2} x-3$$ 2. $$h(x)=x^{2}+3$$ where $$x \geq 0$$ Answer 1. $$f^{-1}(x)=2 x+6$$ Exercise $$\PageIndex{21}$$ Sketch the graph. 1. $$f(x)=e^{x}-5$$ 2. $$g(x)=10^{-x}$$ 3. Joe invested$$$5,200$$ in an account earning $$3.8$$% annual interest that is compounded monthly. How much will be in the account at the end of $$4$$ years?
4. Mary has $$$3,500$$ in a savings account earning $$4 \frac{1}{2}$$% annual interest that is compounded continuously. How much will be in the account at the end of $$3$$ years? Answer 1. Figure 7.E.15 3.$$$6,052.18$$

Exercise $$\PageIndex{22}$$

Evaluate.

1. $$\log _{3} 81$$
2. $$\log _{2}\left(\frac{1}{4}\right)$$
3. $$\log 1,000$$
4. $$\ln e$$
5. $$\log _{4} 2$$
6. $$\log _{9}\left(\frac{1}{3}\right)$$
7. $$\ln e^{3}$$
8. $$\log _{1 / 5} 25$$

1. $$4$$

2. $$-2$$

3. $$3$$

4. $$1$$

Exercise $$\PageIndex{23}$$

Sketch the graph.

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

1.

Exercise $$\PageIndex{24}$$

1. Expand: $$\log \left(\frac{100 x^{2} y}{\sqrt{z}}\right)$$.
2. Write as a single logarithm with coefficient $$1$$: $$2 \log _{2} x+\frac{1}{3} \log _{2} y-3 \log _{2} z$$.

1. $$2+2 \log x+\log y-\frac{1}{2} \log z$$

Exercise $$\PageIndex{25}$$

Evaluate. Round off to the nearest tenth.

1. $$\log _{2} 10$$
2. $$\ln 1$$
3. $$\log _{3}\left(\frac{1}{5}\right)$$

1. $$3.3$$

2. $$0$$

3. $$-1.5$$

Exercise $$\PageIndex{26}$$

Solve:

1. $$2^{3 x-1}=16$$
2. $$3^{7 x+1}=5$$
3. $$\log _{5}(3 x-4)=\log _{5}(2 x+7)$$
4. $$\log _{3}\left(x^{2}+26\right)=3$$
5. $$\log _{2} x+\log _{2}(2 x+7)=2$$
6. $$\log (2 x+3)=1+\log (x+1)$$
7. Joe invested $$$5,200$$ in an account earning $$3.8$$% annual interest that is compounded monthly. How long will it take to accumulate a total of$$$6,200$$ in the account?
8. Mary has \$$$3,500$$ in a savings account earning $$4 \frac{1}{2}$$% annual interest that is compounded continuously. How long will it take to double the amount in the account?
9. During the exponential growth phase, certain bacteria can grow at a rate of $$5.3$$% per hour. If $$12,000$$ cells are initially present in a sample, construct an exponential growth model and use it to:
1. Estimate the population of bacteria in $$3.5$$ hours.
2. Estimate the time it will take the population to double.
10. The half-life of caesium-137 is about $$30$$ years. Approximate the time it will take a $$20$$-milligram sample of caesium-137 to decay to $$8$$ milligrams.
2. $$\frac{\log 5-\log 3}{7 \log 3}$$
4. $$\pm 1$$
6. $$-\frac{7}{8}$$
8. $$15.4$$ years
10. $$40$$ years