Chapter 5

\(55-62\) . Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$ \log _{12} 2.5 $$

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

Use a graphing device to find all solutions of the equation, rounded to two decimal places. \(e^{x}=-x\)

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

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

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?

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

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

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

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

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

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

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

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

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

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

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

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 that \(k=2.1, c=8000\) , 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 S10 million or more?

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

Solve the inequality. \(x^{2} e^{x}-2 e^{x}<0\)

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.

Find the inverse function of \(f\). \(f(x)=2^{2 x}\)

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 $$M=-2.5 \log \left(\frac{B}{B_{0}}\right)$$ where \(B\) is the actual brightness of the star and \(B_{0}\) is a constant. (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 bright than Albiero.

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

Find the inverse function of \(f\). \(f(x)=3^{x+1}\)

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) \(-\ln \left(\frac{1}{A}\right)=\ln A\)

Draw the graph of the function in a suitable viewing rec- tangle, and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=\frac{\ln x}{x} $$

Find the inverse function of \(f\). \(f(x)=\log _{2}(x-1)\)

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

Draw the graph of the function in a suitable viewing rec- tangle, and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=x \log _{10}(x+10) $$

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 \(h(x)=\ln x\) horizontal shrinking is the same as a vertical shift.

Find the functions \(f \circ g\) and \(g \circ f\) and their domains. $$ f(x)=2^{x}, \quad g(x)=x+1 $$

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 functions \(f \circ g\) and \(g \circ f\) and their domains. $$ f(x)=3^{x}, \quad g(x)=x^{2}+1 $$

A woman 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 \(\$ 8000 ?\)

Find the functions \(f \circ g\) and \(g \circ f\) and their domains. $$ f(x)=\log _{2} x, \quad g(x)=x-2 $$

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

Find the functions \(f \circ g\) and \(g \circ f\) and their domains. $$ f(x)=\log x, \quad g(x)=x^{2} $$

Nancy wants to invest \(\$ 4000\) in saving certificates that bear an interest rate of 9.75\(\%\) per year, compounded semiannully. How long a time period should she choose to save an amount of \(\$ 5000 ?\)

Compare the rates of growth of the functions \(f(x)=\ln x\) and \(g(x)=\sqrt{x}\) by drawing their graphs on a common screen using the viewing rectangle \([-1,30]\) by \([-1,6] .\)

How long will it take for an investment of \(\$ 1000\) to double in value if the interest rate is 8.5\(\%\) per year, compounded continuously?

(a) By drawing the graphs of the functions $$ f(x)=1+\ln (1+x) \quad \text { and } \quad g(x)=\sqrt{x} $$ in a suitable viewing rectangle, show that even when a logarithmic function starts out higher than a root function, it is ultimately overtaken by the root function. (b) Find, correct to two decimal places, the solutions of the equation \(\sqrt{x}=1+\ln (1+x)\)

A sum of \(\$ 1000\) was invested for 4 years, and the interest was compounded semiannually. If this sum amounted to \(\$ 1435.77\) in the given time, what was the interest rate?

A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .(\text { b) How are the graphs in part (a) }\) related? $$ f(x)=\log (c x) $$

A 15 -g sample of radioactive iodine decays in such a way that the mass remaining after \(t\) days is given by \(m(t)=15 e^{-0.087 t},\) where \(m(t)\) is measured in grams. After how many days is there only 5 g remaining?

A family of functions is given. (a) Draw graphs of the family for \(c=1,2,3,\) and \(4 .(\text { b) How are the graphs in part (a) }\) related? $$ f(x)=c \log x $$

The velocity of a sky diver \(t\) seconds after jumping is given by \(v(t)=80\left(1-e^{-0.2 t}\right) .\) After how many seconds is the velocity 70 \(\mathrm{ft} / \mathrm{s} ?\)

A function \(f(x)\) is given. (a) Find the domain of the function \(f .\) (b) Find the inverse function of \(f .\) $$ f(x)=\log _{2}\left(\log _{10} x\right) $$

A small lake is stocked with a certain species of fish. The fish population is modeled by the function $$P=\frac{10}{1+4 e^{-0.8 t}}$$ where \(P\) is the number of fish in thousands and \(t\) is measured in years since the lake was stocked. (a) Find the fish population after 3 years. (b) After how many years will the fish population reach 5000 fish?

A function \(f(x)\) is given. (a) Find the domain of the function \(f .\) (b) Find the inverse function of \(f .\) $$ f(x)=\ln (\ln (\ln x)) $$