Chapter 4

In Exercises \(47-52,\) let \(b=\log\) k. Write each expression in terms of b. Assume \(k>0\). $$\log 100 k$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$. \log _{7} 150$$

Explain why the function \(f(t)=e^{(1 / 2) t}\) cannot model exponential decay.

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-4 x^{5}+9$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln (2 x)=1+\ln (x+3)$$

In Exercises \(47-52,\) let \(b=\log\) k. Write each expression in terms of b. Assume \(k>0\). $$\log k^{3}$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=2 x^{5}-6$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{3} x=2+\log _{3}(x-2)$$

In Exercises \(47-52,\) let \(b=\log\) k. Write each expression in terms of b. Assume \(k>0\). $$\log k^{4}$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{7} 230$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{1}{x}$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (3 x+1)-\log \left(x^{2}+1\right)=0$$

In Exercises \(47-52,\) let \(b=\log\) k. Write each expression in terms of b. Assume \(k>0\). $$\log \frac{1}{k}$$

Use the definition of a logarithm to solve for \(x\). $$\log _{2} x=3$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=\frac{-1}{2 x}$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (x+5)-\log \left(4 x^{2}+5\right)=0$$

In Exercises \(47-52,\) let \(b=\log\) k. Write each expression in terms of b. Assume \(k>0\). $$\log \frac{1}{k^{3}}$$

Use the definition of a logarithm to solve for \(x\). $$ \log _{5} \sqrt{5}=x$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=(x-1)^{2}, x \geq 1$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (2 x+5)+\log (x+1)=1$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log 10^{\sqrt{2}}$$

Use a graphing utility to solve each equation for \(x.\) $$5=3^{x}$$

Use the definition of a logarithm to solve for \(x\). $$\log _{3} x=\frac{1}{3}$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=(x+2)^{2}, x \geq-2$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (3 x+1)+\log (x+1)=1$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log 10^{2 x}$$

Use a graphing utility to solve each equation for \(x.\) $$7=4^{x}$$

Use the definition of a logarithm to solve for \(x\). $$\log _{6} x=-2$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\sqrt{x+3}, x \geq-3$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{2}(x+5)=\log _{2}(x)+\log _{2}(x-3)$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\ln e^{\sqrt{3}}$$

Use a graphing utility to solve each equation for \(x.\) $$10=2^{-x}$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\sqrt{x-4}, x \geq 4$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln 2 x-\ln \left(x^{2}+1\right)=\ln 1$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\ln e^{(x+1)}$$

Use a graphing utility to solve each equation for \(x.\) $$20=100(5)^{-x}$$

Use the definition of a logarithm to solve for \(x\). $$\log _{x} 9=\frac{1}{2}$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{2 x}{x-1}$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$2 \ln x+\ln (x-1)=3.1$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$10^{\log (5 x)}$$

Use a graphing utility to solve each equation for \(x.\) $$100=50 e^{0.06 x}$$

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=2 \log x$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{x+3}{x}$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$-\ln x-\ln (x+2)=2.5$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$e^{\ln \left(5 x^{2}-1\right)}$$

Use a graphing utility to solve each equation for \(x.\) $$25=50 e^{-0.05 x}$$

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=4 \ln x$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log |x-2|+\log |x|=1.2$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$10^{\log (3 x+1)}$$

Consider the function \(f(x)=x e^{-x}.\) (a) Use a graphing utility to graph this function, with \(x\) ranging from -5 to \(5 .\) You may need to scroll through the table of values to set an appropriate scale for the vertical axis. (b) What are the domain and range of \(f ?\) (c) What are the \(x\) - and \(y\) -intercepts, if any, of the graph of this function? (d) Describe the behavior of the function as \(x\) approaches \(\pm \infty.\)