Chapter 4

Think About It Graph \(y=3^{x}\) and \(y=4^{x}\). Use the graph to solve the inequality \(3^{x} < 4^{x}\)

Use the properties of natural logarithms to rewrite the expression. $$\ln \frac{1}{e^{4}}$$

(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.$$y_{1}=\frac{1}{2} \ln x-\ln (x+2), \quad y_{2}=\ln \left(\frac{\sqrt{x}}{x+2}\right)$$

Use the zero or root feature or the zoom and trace features of a graphing utility to approximate the solution of the exponential equation accurate to three decimal places. $$\frac{119}{e^{6 x}-14}=7$$

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. Verify using a graphing utility. $$f(x)=\ln (x-1)$$

Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$g(x)=6 e^{1-x}-25$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{3} 9$$.

Think About It In Exercises \(89-92,\) place the correct symbol \(( < \text { or } > )\) between the two numbers. $$e^{\pi} \quad \pi^{e}$$

Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$f(x)=3 e^{3 x / 2}-962$$

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. Verify using a graphing utility. $$h(x)=\ln (x+1)$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{6} 6$$.

Think About It In Exercises \(89-92,\) place the correct symbol \(( < \text { or } > )\) between the two numbers. $$2^{10} \quad 10^{2}$$

Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$g(t)=e^{0.09 t}-3$$

Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. Verify using a graphing utility. $$g(x)=\ln (-x)$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{4} 16^{3.4}$$.

Think About It In Exercises \(89-92,\) place the correct symbol \(( < \text { or } > )\) between the two numbers. $$5^{-3} \quad 3^{-5}$$

Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$h(t)=e^{-0.125 t}-8$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{5}\left(\frac{1}{125}\right)$$.

Think About It In Exercises \(89-92,\) place the correct symbol \(( < \text { or } > )\) between the two numbers. $$4^{1 / 2} \quad\left(\frac{1}{2}\right)^{4}$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln x=-3$$

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln (x+8)$$

Determine whether the function has an inverse function. If it does, find \(f^{-1}\). $$f(x)=5 x-7$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln x=-4$$

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln (x-4)$$

Determine whether the function has an inverse function. If it does, find \(f^{-1}\). $$f(x)=-\frac{2}{3} x+\frac{5}{2}$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln 4 x=2.1$$

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln x-5$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{5} 375-\log _{5} 3$$.

Determine whether the function has an inverse function. If it does, find \(f^{-1}\). $$f(x)=\sqrt[3]{x+8}$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln 2 x=1.5$$

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln x+4$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{4} 2+\log _{4} 32$$

Determine whether the function has an inverse function. If it does, find \(f^{-1}\). $$f(x)=\sqrt{x^{2}+6}$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{5}(3 x+2)=\log _{5}(-x)$$

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln (x-1)+2$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln e^{3}-\ln e^{7}$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{9}(4+x)=\log _{9} 2 x$$

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln (x+2)-5$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln e^{6}-2 \ln e^{7}$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$-2+2 \ln 3 x=17$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\frac{x}{2}-\ln \frac{x}{4}$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$2 \ln e^{4}$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$3+2 \ln x=10$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$g(x)=6 x \ln x$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$3 \ln e^{5}$$.

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$h(x)=\frac{14 \ln x}{x}$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln \frac{1}{\sqrt{e}}$$.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$4 \log _{10}(x-6)=11$$

(a) use a graphing utility to graph the function, (b) find the domain, (c) use the graph to find the open intervals on which the function is increasing and decreasing, and (d) approximate any relative maximum or minimum values of the function. Round your results to three decimal places. $$f(x)=\frac{x}{\ln x}$$

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln \sqrt[5]{e^{3}}$$.