Chapter 4

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. $$\log (3 x-3)=\log (x+1)+\log 4$$

Find the domain of each logarithmic function. $$f(x)=\ln (x-7)^{2}$$

Use a graphing utility and the change-of-base property to graph each function. \(y=\log _{15} x\)

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. $$\log (2 x-1)=\log (x+3)+\log 3$$

Evaluate or simplify each expression without using a calculator. $$\log 100$$

Use a graphing utility and the change-of-base property to graph each function. \(y=\log _{2}(x+2)\)

You have \(\$ 10,000\) to invest. One bank pays \(5 \%\) interest compounded quarterly and a second bank pays \(4.5 \%\) interest compounded monthly. a. Use the formula for compound interest to write a function for the balance in each bank at any time \(t\) 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. $$2 \log x=\log 25$$

Each group member should consult an almanac, newspaper. magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?

Evaluate or simplify each expression without using a calculator. $$\log 1000$$

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.

Use a graphing utility and the change-of-base property to graph each function. \(y=\log _{3}(x-2)\)

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. $$3 \log x=\log 125$$

Exercises \(83-85\) will help you prepare for the material covered in the first section of the next chapter. a. Does \((4,-1)\) satisfy \(x+2 y=2 ?\) b. Does \((4,-1)\) satisfy \(x-2 y=6 ?\)

Evaluate or simplify each expression without using a calculator. $$\log 10^{7}$$

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

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

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. $$\log (x+4)-\log 2=\log (5 x+1)$$

Exercises \(83-85\) will help you prepare for the material covered in the first section of the next chapter. Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect?

Evaluate or simplify each expression without using a calculator. $$\log 10^{8}$$

Determine whether each statement makes sense or does not make sense, and explain 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.

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

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. $$\log (x+7)-\log 3=\log (7 x+1)$$

Graph \(x+2 y=2\) and \(x-2 y=6\) in the same rectangular coordinate system. At what point do the graphs intersect? $$ \text { Solve: } 5(2 x-3)-4 x=9 $$

Evaluate or simplify each expression without using a calculator. $$10^{\log 33}$$

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

Evaluate or simplify each expression without using a calculator. $$10^{\log 53}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I use the natural base \(e\) when determining how much money I'd have in a bank account that earns compound interest subject to continuous compounding.

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

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. $$\log (x-2)+\log 5=\log 100$$

Evaluate or simplify each expression without using a calculator. $$\ln 1$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. As the number of compounding periods increases on a fixed investment, the amount of money in the account over a fixed interval of time will increase without bound.

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

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. $$\log x+\log (x+3)=\log 10$$

Evaluate or simplify each expression without using a calculator. $$\ln e$$

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

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. $$\log (x+3)+\log (x-2)=\log 14$$

Evaluate or simplify each expression without using a calculator. $$\ln e^{6}$$

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. $$\ln (x-4)+\ln (x+1)=\ln (x-8)$$

Evaluate or simplify each expression without using a calculator. $$\ln e^{7}$$

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.

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. $$\log _{2}(x-1)-\log _{2}(x+3)=\log _{2}\left(\frac{1}{x}\right)$$

Evaluate or simplify each expression without using a calculator. $$\ln \frac{1}{e^{6}}$$

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. $$\ln (x-2)-\ln (x+3)=\ln (x-1)-\ln (x+7)$$

Evaluate or simplify each expression without using a calculator. $$\ln \frac{1}{e^{7}}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

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. $$\ln (x-5)-\ln (x+4)=\ln (x-1)-\ln (x+2)$$

Evaluate or simplify each expression without using a calculator. $$e^{\ln 125}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The hyperbolic cosine and hyperbolic sine functions are defined by $$\cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2}$$ a. Show that \(\cosh x\) is an even function. b. Show that \(\sinh x\) is an odd function. c. Prove that \((\cosh x)^{2}-(\sinh x)^{2}=1\)

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. \(x \log 10^{x}=x^{2}\)