Chapter 4

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln e=1$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$7-2 e^{x}=1$$

(a) use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values. $$f(x)=x^{3} e^{x}$$

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{b} \frac{\sqrt{x} y^{4}}{z^{4}}$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln e^{3}=3$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$-14+3 e^{x}=11$$

(a) use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values. $$f(x)=x^{3} e^{-x+2}$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$\begin{array}{l}y_{1}=\ln \left[x^{2}(x-4)\right] \\\y_{2}=2 \ln x+\ln (x-4)\end{array}$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln \sqrt{e}=\frac{1}{2}$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$5\left(2^{3-x}\right)-13=100$$

Complete the table to determine the balance \(A\) for \(\$ 2500\) invested at rate \(r\) for \(t\) years and compounded \(n\) times per year. $$\begin{array}{|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 12 & 365 & \text { Continuous } \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=2 \%, t=10 \text { years }$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$\begin{array}{l}y_{1}=\ln 9 x^{3} \\\y_{2}=\ln 9+3 \ln x\end{array}$$

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln \frac{1}{e^{2}}=-2$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$6\left(8^{-2-x}\right)+15=2601$$

Complete the table to determine the balance \(A\) for \(\$ 2500\) invested at rate \(r\) for \(t\) years and compounded \(n\) times per year. $$\begin{array}{|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 12 & 365 & \text { Continuous } \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=6 \%, t=10 \text { years }$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$\begin{aligned}&y_{1}=\ln \left(\frac{x^{4}}{x-2}\right)\\\&y_{2}=4 \ln x-\ln (x-2)\end{aligned}$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln 9=2.1972 . . .$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$\left(1+\frac{0.10}{12}\right)^{12 t}=2$$

Complete the table to determine the balance \(A\) for \(\$ 2500\) invested at rate \(r\) for \(t\) years and compounded \(n\) times per year. $$\begin{array}{|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 12 & 365 & \text { Continuous } \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=4 \%, t=20 \text { years }$$

(a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically.$$\begin{array}{l}y_{1}=\ln \left(\frac{\sqrt{x}}{x+3}\right) \\\y_{2}=\frac{1}{2} \ln x-\ln (x+3)\end{array}$$.

Write the logarithmic equation in exponential form. For example, the exponential form of \(\ln 5=1.6094\). . . is \(e^{1.6094 \cdots}=5.\) $$\ln \sqrt[3]{e}=\frac{1}{3}$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$\left(16+\frac{0.878}{26}\right)^{3 t}=30$$

Complete the table to determine the balance \(A\) for \(\$ 2500\) invested at rate \(r\) for \(t\) years and compounded \(n\) times per year. $$\begin{array}{|c|c|c|c|c|c|c|} \hline n & 1 & 2 & 4 & 12 & 365 & \text { Continuous } \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=3 \%, t=40 \text { years }$$

Use the properties of logarithms to condense the expression.$$\ln x+\ln 4$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{3}=20.0855 . . . $$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$5000\left[\frac{(1+0.005)^{x}}{0.005}\right]=250,000$$

Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & 1 & 10 & 20 & 30 & 40 & 50 \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=4 \%$$

Use the properties of logarithms to condense the expression.$$\ln y+\ln z$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{0}=1$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$250\left[\frac{(1+0.01)^{x}}{0.01}\right]=150,000$$

Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & 1 & 10 & 20 & 30 & 40 & 50 \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=6 \%$$

Use the properties of logarithms to condense the expression.$$\log _{4} z-\log _{4} y$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{1.3}=3.6692 . . .$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$e^{2 x}-4 e^{x}-5=0$$

Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & 1 & 10 & 20 & 30 & 40 & 50 \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=3.5 \%$$

Use the properties of logarithms to condense the expression.$$\log _{5} 8-\log _{5} t$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{2.5}=12.1824 . . .$$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$e^{2 x}-e^{x}-6=0$$

Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. $$\begin{array}{|c|c|c|c|c|c|c|} \hline t & 1 & 10 & 20 & 30 & 40 & 50 \\ \hline A & & & & & & \\ \hline \end{array}$$ $$r=2.5 \%$$

Use the Leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial function. $$f(x)=2 x^{3}-3 x^{2}+x-1$$

Use the properties of logarithms to condense the expression.$$4 \log _{3}(x+2)$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$\sqrt[3]{e}=1.3956 . . . $$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$e^{x}=e^{x^{2}-2}$$

You build an annuity by investing \(P\) dollars every month at interest rate \(r,\) compounded monthly. Find the amount \(A\) accrued after \(n\) months using the formula. \(A=P\left[\frac{(1+r / 12)^{n}-1}{r / 12}\right],\) where \(r\) is in decimal form. $$P=\mathrm{S} 25, r=0.12, n=48 \mathrm{months}$$

Use the Leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial function. $$f(x)=5-x^{2}-4 x^{4}$$

Use the properties of logarithms to condense the expression.$$\frac{5}{2} \log _{7}(z-4)$$.

Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$\frac{1}{e^{4}}=0.0183. . . $$

Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$e^{2 x}=e^{x^{2}-8}$$

You build an annuity by investing \(P\) dollars every month at interest rate \(r,\) compounded monthly. Find the amount \(A\) accrued after \(n\) months using the formula. \(A=P\left[\frac{(1+r / 12)^{n}-1}{r / 12}\right],\) where \(r\) is in decimal form. $$P=\$ 100, r=0.09, n=60 \text { months }$$

Use the Leading Coefficient Test to determine the right-hand and left-hand behavior of the graph of the polynomial function. $$g(x)=-1.6 x^{5}+4 x^{2}-2$$