Chapter 10

Approximate each logarithm to three decimal places. $$ \log _{5} 0.26 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=-3.3244 $$

Evaluate each logarithmic expression. $$ \log _{10}\left(\log _{7} 7\right) $$

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=-3 x-4 $$

For Problems \(39-44\), graph each of the exponential functions. See answer section. $$ f(x)=e^{x}+1 $$

Graph each of the exponential functions. $$ f(x)=2^{-x-2} $$

Approximate each logarithm to three decimal places. $$ \log _{5} 0.047 $$

Use your calculator to find \(x\) when given \(\ln x\). Express answers to five significant digits. $$ \ln x=-2.3745 $$

Evaluate each logarithmic expression. $$ \log _{2}\left(\log _{5} 5\right) $$

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=-5 x+6 $$

Graph each of the exponential functions. See answer section. $$ f(x)=e^{x}-2 $$

Graph each of the exponential functions. $$ f(x)=2^{-x+1} $$

Approximate each logarithm to three decimal places. $$ \log _{7} 500 $$

(a) Complete the following table, and then graph \(f(x)=\log x\). (Express the values for \(\log x\) to the nearest tenth.) $$ \begin{array}{c|c|c|c|c|c|c|c} \hline \boldsymbol{x} & 0.1 & 0.5 & 1 & 2 & 4 & 8 & 10 \\ \hline \log \boldsymbol{x} & & & & & & & \\ \hline \end{array} $$ (b) Complete the following table, expressing values for \(10^{x}\) to the nearest tenth. $$ \begin{array}{c|c|c|c|c|c|c|c} \hline \boldsymbol{x} & -1 & -0.3 & 0 & 0.3 & 0.6 & 0.9 & 1 \\ \hline 10^{\boldsymbol{x}} & & & & & & & \\ \hline \end{array} $$ Then graph \(f(x)=10^{x}\), and reflect it across the line \(y=\) \(x\) to produce the graph for \(f(x)=\log x\).

Solve each equation. \(\log _{7} x=2\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=\frac{3}{4} x-\frac{5}{6} $$

Graph each of the exponential functions. $$ f(x)=2^{x^{2}} $$

Approximate each logarithm to three decimal places. $$ \log _{8} 750 $$

(a) Complete the following table, and then graph \(f(x)=\ln x\). (Express the values for \(\ln x\) to the nearest tenth.) $$ \begin{array}{c|l|l|l|l|l|l|l} \hline \boldsymbol{x} & 0.1 & 0.5 & 1 & 2 & 4 & 8 & 10 \\ \hline \ln \boldsymbol{x} & & & & & & & \\ \hline \end{array} $$ (b) Complete the following table, expressing values for \(e^{x}\) to the nearest tenth. \begin{array}{c|c|c|c|c|c|c|c} \hline \boldsymbol{x} & -2.3 & -0.7 & 0 & 0.7 & 1.4 & 2.1 & 2.3 \\ \hline \boldsymbol{e}^{\boldsymbol{x}} & & & & & & & \\ \hline \end{array} Then graph \(f(x)=e^{x}\), and reflect it across the line \(y=\) \(x\) to produce the graph for \(f(x)=\ln x\).

Solve each equation. \(\log _{2} x=5\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=\frac{2}{3} x-\frac{1}{4} $$

Graph each of the exponential functions. See answer section. $$ f(x)=-e^{x} $$

Graph each of the exponential functions. $$ f(x)=2^{x}+2^{-x} $$

How long will it take \(\$ 750\) to be worth \(\$ 1000\) if it is invested at \(12 \%\) interest compounded quarterly?

Solve each equation. \(\log _{8} x=\frac{4}{3}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=-\frac{2}{3} x $$

Graph each of the exponential functions. See answer section. $$ f(x)=e^{2 x} $$ 4

Graph each of the exponential functions. $$ f(x)=2^{|x|} $$

How long will it take \(\$ 1000\) to double if it is invested at \(9 \%\) interest compounded semiannually?

Graph \(y=\log _{2} x\) by graphing \(2^{y}=x\)

Solve each equation. \(\log _{16} x=\frac{3}{2}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=\frac{4}{3} x $$

Graph each of the exponential functions. See answer section. $$ f(x)=e^{-x} $$

Graph each of the exponential functions. $$ f(x)=3^{1-x^{2}} $$

How long will it take \(\$ 2000\) to double if it is invested at \(13 \%\) interest compounded continuously?

Graph \(f(x)=\log _{3} x\) by reflecting the graph of \(g(x)=3^{x}\) across the line \(y=x\).

Solve each equation. \(\log _{9} x=\frac{3}{2}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=\sqrt{x} \quad \text { for } x \geq 0 $$

Explain the difference between simple interest and compound interest.

Graph each of the exponential functions. $$ f(x)=2^{x}-2^{-x} $$

How long will it take \(\$ 500\) to triple if it is invested at \(9 \%\) interest compounded continuously? \(12.2\) years

Graph \(f(x)=\log _{4} x\) by reflecting the graph of \(g(x)=4^{x}\) across the line \(y=x\).

Solve each equation. \(\log _{8} x=-\frac{2}{3}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=\frac{1}{x} \quad \text { for } x \neq 0 $$

Would it be better to invest \(\$ 5000\) at \(6.25 \%\) interest compounded annually for 5 years or to invest \(\$ 5000\) at \(6.25 \%\) interest compounded continuously for 5 years? Explain your answer.

Graph each of the exponential functions. $$ f(x)=2^{-|x|} $$

What rate of interest compounded continuously is needed for an investment of \(\$ 500\) to grow to \(\$ 900\) in 10 years?

Solve each equation. \(\log _{4} x=-\frac{3}{2}\)

(a) find \(f^{-1}\) and (b) verify that \(\left(f \circ f^{-1}\right)(x)=x\) and \(\left(f^{-1} \circ f\right)(x)=x\). $$ f(x)=x^{2}+4 \quad \text { for } x \geq 0 $$

How would you explain the concept of effective yield to someone who missed class when it was discussed?