Chapter 5

Illustrate the definition of the number \(e\) by graphing the curve \(y=(1+1 / x)^{x}\) and the line \(y=e\) on the same screen using the viewing rectangle \([0,40]\) by \([0,4].\)

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ \log x=x^{2}-2 $$

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{12} 2.5 $$

Investigate the behavior of the function $$ f(x)=\left(1-\frac{1}{x}\right)^{x} $$ as \(x \rightarrow \infty\) by graphing \(f\) and the line \(y=1 / e\) on the same screen using the viewing rectangle \([0,20]\) by \([0,1].\)

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ x^{3}-x=\log (x+1) $$

Use the Change of Base Formula to show that $$ \log _{3} x=\frac{\ln x}{\ln 3} $$ Then use this fact to draw the graph of the function \(f(x)=\log _{3} x .\)

(a) Draw the graphs of the family of functions $$ f(x)=\frac{a}{2}\left(e^{x / a}+e^{-x / a}\right) $$ (b) How does a larger value of \(a\) affect the graph?

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ x=\ln \left(4-x^{2}\right) $$

Draw graphs of the family of functions \(y=\log _{a} x\) for \(a=2, e, 5,\) and 10 on the same screen, using the viewing rectangle \([0,5]\) by \([-3,3] .\) How are these graphs related?

58–59 ? Graph the function and comment on vertical and horizontal asymptotes. $$y=2^{1 / x}$$

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ e^{x}=-x $$

Use the Change of Base Formula to show that $$ \log e=\frac{1}{\ln 10} $$

\(59-64\) Find the domain of the function. $$ f(x)=\log _{10}(x+3) $$

58–59 ? Graph the function and comment on vertical and horizontal asymptotes. $$ y=\frac{e^{x}}{x} $$

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ 2^{-x}=x-1 $$

Simplify: \(\left(\log _{2} 5\right)\left(\log _{5} 7\right)\)

\(59-64\) Find the domain of the function. $$ f(x)=\log _{5}(8-2 x) $$

60–61 ? Find the local maximum and minimum values of the function and the value of x at which each occurs. State each answer correct to two decimal places. $$ g(x)=x^{x} \quad(x>0) $$

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ 4^{-x}=\sqrt{x} $$

Show that \(-\ln \left(x-\sqrt{x^{2}-1}\right)=\ln \left(x+\sqrt{x^{2}-1}\right)\)

\(59-64\) Find the domain of the function. $$ g(x)=\log _{3}\left(x^{2}-1\right) $$

60–61 ? Find the local maximum and minimum values of the function and the value of x at which each occurs. State each answer correct to two decimal places. $$ g(x)=e^{x}+e^{-3 x} $$

Use a graphing device to find all solutions of the equation, correct to two decimal places. $$ e^{x^{2}}-2=x^{3}-x $$

Forgetting Use the Ebbinghaus Forgetting Law (Example 5) to estimate a student's score on a biology test two years after he got a score of 80 on a test covering the same material. Assume \(c=0.3\) and \(t\) is measured in months.

\(59-64\) Find the domain of the function. $$ g(x)=\ln \left(x-x^{2}\right) $$

62–63 ? Find, correct to two decimal places, (a) the intervals on which the function is increasing or decreasing, and (b) the range of the function. $$ y=10^{x-x^{2}} $$

Solve the inequality. $$ \log (x-2)+\log (9-x)<1 $$

Wealth Distribution Vilfredo Pareto \((1848-1923)\) observed that most of the wealth of a country is owned by a few members of the population. Pareto's Principle is $$ \log P=\log c-k \log W $$ where \(W\) is the wealth level (how much money a person has) and \(P\) is the number of people in the population having that much money. (a) Solve the equation for \(P\) (b) Assume \(k=2.1, c=8000\) , and \(W\) is measured in millions of dollars. Use part (a) to find the number of people who have \(\$ 2\) million or more. How many people have \(\$ 10\) million or more?

\(59-64\) Find the domain of the function. $$ h(x)=\ln x+\ln (2-x) $$

62–63 ? Find, correct to two decimal places, (a) the intervals on which the function is increasing or decreasing, and (b) the range of the function. $$ y=x e^{-x} $$

Solve the inequality. $$ 3 \leq \log _{2} x \leq 4 $$

Biodiversity Some biologists model the number of species \(S\) in a fixed area \(A\) (such as an island) by the Species-Area relationship $$ \log S=\log c+k \log A $$ where \(c\) and \(k\) are positive constants that depend on the type of species and habitat. (a) Solve the equation for \(S\) (b) Use part (a) to show that if \(k=3\) then doubling the area increases the number of species eightfold.

\(59-64\) Find the domain of the function. $$ h(x)=\sqrt{x-2}-\log _{5}(10-x) $$

Medical Drugs When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient’s bloodstream after \(t\) hours is modeled by $$ D(t)=50 e^{-0.2 t} $$ How many milligrams of the drug remain in the patient's bloodstream after 3 hours?

Magnitude of Stars The magnitude M of a star is a measure of how bright a star appears to the human eye. It is defined by $$ M=-2.5 \log \left(\frac{B}{B_{0}}\right) $$ (a) Expand the right-hand side of the equation. (b) Use part (a) to show that the brighter a star, the less its magnitude. (c) Betelgeuse is about 100 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less than Albiero.

\(65-70\) Draw the graph of the function in a suitable viewing rectangle and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=\log _{10}\left(1-x^{2}\right) $$

Radioactive Decay A radioactive substance decays in such a way that the amount of mass remaining after \(t\) days is given by the function $$ m(t)=13 e^{-0.015 t} $$ where \(m(t)\) is measured in kilograms. (a) Find the mass at time \(t=0\) . (b) How much of the mass remains after 45 days?

Solve the inequality. $$ x^{2} e^{x}-2 e^{x}<0 $$

True or False? Discuss each equation and determine whether it is true for all possible values of the variables. (Ignore values of the variables for which any term is undefined.) (a) \(\log \left(\frac{x}{y}\right)=\frac{\log x}{\log y}\) (b) \(\log _{2}(x-y)=\log _{2} x-\log _{2} y\) (c) \(\log _{5}\left(\frac{a}{b^{2}}\right)=\log _{5} a-2 \log _{5} b\) (d) \(\log 2^{z}=z \log 2\) (e) \((\log P)(\log Q)=\log P+\log Q\) (f) \(\frac{\log a}{\log b}=\log a-\log b\) (g) \(\left(\log _{2} 7\right)^{x}=x \log _{2} 7\) (h) \(\log _{a} a^{a}=a\) (i) \(\log (x-y)=\frac{\log x}{\log y}\) (j) \(-\ln \left(\frac{1}{A}\right)=\ln A\)

Radioactive Decay Radioactive iodine is used by doctors as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after \(t\) days is given by the function $$ m(t)=6 e^{-0.087 t} $$ where \(m(t)\) is measured in grams. (a) Find the mass at time \(t=0\) (b) How much of the mass remains after 20 days?

Compound Interest \(\quad\) A man invests \(\$ 5000\) in an account that pays 8.5\(\%\) interest per year, compounded quarterly. (a) Find the amount after 3 years. (b) How long will it take for the investment to double?

Find the Error \(\quad\) What is wrong with the following argument? $$ \begin{aligned} \log 0.1 &<2 \log 0.1 \\ &=\log (0.1)^{2} \\ &=\log 0.01 \\\ \log 0.1 &<\log 0.01 \\ 0.1 &<0.01 \end{aligned} $$

\(65-70\) Draw the graph of the function in a suitable viewing rectangle and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=x+\ln x $$

Sky Diving A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is \(0.2 .\) It can be shown that the downward velocity of the sky diver at time \(t\) is given by $$ v(t)=80\left(1-e^{-0.2 t}\right) $$ where \(t\) is measured in seconds and \(v(t)\) is measured in feet per second \((\mathrm{ft} / \mathrm{s})\) (a) Find the initial velocity of the sky diver. (b) Find the velocity after 5 s and after 10 s. (c) Draw a graph of the velocity function \(v(t)\) . (d) The maximum velocity of a falling object with wind resistance is called its terminal velocity. From the graph in part (c) find the terminal velocity of this sky diver.

Compound Interest \(\quad\) A man invests \(\$ 6500\) in an account that pays 6\(\%\) interest per year, compounded continuously. (a) What is the amount after 2 years? (b) How long will it take for the amount to be \(\$ 8000 ?\)

Shifting, Shrinking, and Stretching Graphs of Functions Let \(f(x)=x^{2} .\) Show that \(f(2 x)=4 f(x),\) and explain how this shows that shrinking the graph of \(f\) horizontally has the same effect as stretching it vertically. Then use the identities \(e^{2+x}=e^{2} e^{x}\) and \(\ln (2 x)=\ln 2+\ln x\) to show that for \(g(x)=e^{x},\) a horizontal shift is the same as a vertical stretch and for \(h(x)=\ln x,\) a horizontal shrinking is the same as a vertical shift.

\(65-70\) Draw the graph of the function in a suitable viewing rectangle and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=x(\ln x)^{2} $$

Mixtures and Concentrations \(A 50\) -gallon barrel is filled completely with pure water. Salt water with a concentration of 0.3 Ib/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time \(t\) is given by $$ Q(t)=15\left(1-e^{-0.04 t}\right) $$ where \(t\) is measured in minutes and \(Q(t)\) is measured in pounds. (a) How much salt is in the barrel after 5 \(\mathrm{min} ?\) (b) How much salt is in the barrel after 10 \(\mathrm{min}\) ? (c) Draw a graph of the function \(Q(t)\) (d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as \(t\) becomes large. Is this what you would expect?

Compound Interest Find the time required for an investment of \(\$ 5000\) to grow to \(\$ 8000\) at an interest rate of 7.5\(\%\) per year, compounded quarterly.

Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model $$ P(t)=\frac{d}{1+k e^{-c t}} $$ where \(c, d,\) and \(k\) arc positive constants. For a certain fish population in a small pond \(d=1200, k=11, c=0.2,\) and \(t\) is measured in years. The fish were introduced into the pond at time \(t=0\) . (a) How many fish were originally put in the pond? (b) Find the population after \(10,20,\) and 30 years. (c) Evaluate \(P(t)\) for large values of \(t\) . What value does the population approach as \(t \rightarrow \infty\) Does the graph shown confirm your calculations?