Chapter 12

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{\pi} 400$$

Simplify each expression. $$\ln e^{13 x}$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{2}(x+2)-\log _{2}(x-5)=3$$

Simplify each expression. $$e^{\ln 5 x^{2}}$$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} \frac{3}{2}$$

You have \(\$ 10,000\) to invest. One bank pays \(5 \%\) interest compounded quarterly and the other pays \(4.5 \%\) interest compounded monthly. a. Use the formula for compound interest to write a function for the balance in each account at any time \(t\) in years. b. Use a graphing utility to graph both functions in an appropriate viewing rectangle. Based on the graphs, which bank offers the better return on your money?

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{4}(x+2)-\log _{4}(x-1)=1$$

Simplify each expression. $$e^{\ln 7 x^{2}}$$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} 6$$

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts (a)-(c). Try generalizing this observation.

Simplify each expression. $$10^{\log \sqrt{x}}$$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} 8$$

Determine whether each statement "makes sense" or "does not make sense" and explair= your reasoning. My graph of \(f(x)=3 \cdot 2^{x}\) shows that the horizontal asymptote for \(f\) is \(x=3\)

Simplify each expression. $$10^{\log \sqrt[3]{x}}$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (2 x-1)-\log x=2$$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} 81$$

Determine whether each statement "makes sense" or "does not make sense" and explair= your reasoning. I'm using a photocopier to reduce an image over and over by \(50 \%,\) so the exponential function \(f(x)=\left(\frac{1}{2}\right)^{x}\) models the new image size, where \(x\) is the number of reductions.

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{3}(x-1)=2$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln (x+1)-\ln x=1$$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} \sqrt{\frac{2}{27}}$$

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{5}(x+4)=2$$

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). $$\log _{b} \sqrt{\frac{3}{16}}$$

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{4} x=-3$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{3}(x+4)=\log _{3} 7$$

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{64} x=\frac{2}{3}$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{2}(x-5)=\log _{2} 4$$

Evaluate each expression without using a calculator. $$\log _{3}\left(\log _{7} 7\right)$$

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{4}\left(2 x^{3}\right)=3 \log _{4}(2 x)$$

Evaluate each expression without using a calculator. $$\log _{5}\left(\log _{2} 32\right)$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (5 x+1)=\log (2 x+3)+\log 2$$

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\ln \left(8 x^{3}\right)=3 \ln (2 x)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The functions \(f(x)=\left(\frac{1}{3}\right)^{x}\) and \(g(x)=3^{-x}\) have the same graph.

Evaluate each expression without using a calculator. $$\log _{2}\left(\log _{3} 81\right)$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (3 x-3)=\log (x+1)+\log 4$$

Evaluate each expression without using a calculator. $$\log (\ln e)$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (2 x-1)=\log (x+3)+\log 3$$

The hyperbolic cosine and hyperbolic sine functions are -defined by $$\cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2}$$ Prove that \((\cosh x)^{2}-(\sinh x)^{2}=1\)

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$2 \log x=\log 25$$

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\ln (5 x)+\ln 1=\ln (5 x)$$

$$\text { Subtract: } \frac{2 x+3}{x^{2}-7 x+12}-\frac{2}{x-3}$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (x+4)-\log 2=\log (5 x+1)$$

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log (x+3)-\log (2 x)=\frac{\log (x+3)}{\log (2 x)}$$

$$\text { Solve: } x(x-3)=10$$

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (x+7)-\log 3=\log (7 x+1)$$

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\log (x+2)}{\log (x-1)}=\log (x+2)-\log (x-1)$$

Will help you prepare for the material covered in the next section. In Section \(8.4,\) we used a switch-and-solve strategy for finding a function's inverse. (See the box on page 625 .) What problem do you encounter when using this strategy to find the inverse of \(f(x)=2^{x} ?\)

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{6}\left(\frac{x-1}{x^{2}+4}\right)=\log _{6}(x-1)-\log _{6}\left(x^{2}+4\right)$$

Will help you prepare for the material covered in the next section. 25 to what power gives \(5 ? \quad\left(25^{?}=5\right)\)

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (x-2)+\log 5=\log 100$$

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{6}[4(x+1)]=\log _{6} 4+\log _{6}(x+1)$$