Chapter 4

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{7} 2.61$$

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

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{6} 532$$

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

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{4} 125$$

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

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 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 \(f(x)=\log _{3} x\)

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 \(-\ln (x-\sqrt{x^{2}-1})=\ln (x+\sqrt{x^{2}-1})\).

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

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

Use the Law of Forgetting (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 that \(c=0.3\) and \(t\) is measured in months.

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. $$y=\log _{10}\left(1-x^{2}\right)$$

Solve the inequality. $$2<10^{x}<5$$

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 \(\$ 10\) million or more?

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. $$y=\ln \left(x^{2}-x\right)$$

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

Some biologists model the number of species \(S\) in a fixed area \(A\) (such as an island) by the speciesarea 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.

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. $$y=x+\ln x$$

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

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.

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

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) \(\log 2^{2}=z \log 2\) (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 rectangle, 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 \(f\). $$f(x)=\log _{2}(x-1)$$

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

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.

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

Compound Interest 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. $$f(x)=3^{x}, \quad g(x)=x^{2}+1$$

Compound Interest 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. $$f(x)=\log _{2} x, \quad g(x)=x-2$$

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.

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

Compound Interest Nancy wants to invest \(\$ 4000\) in saving certificates that bear an interest rate of \(9.75 \%\) per year, compounded semiannually. 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 \([-1,6]\).

Doubling an Investment 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?

Interest Rate \(A\) 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?