Chapter 4

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 4 \ln (x+6)-3 \ln x $$

What is the natural exponential function?

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 8 \ln (x+9)-4 \ln x $$

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x=10,100\) \(1000,10,000,100,000,\) and \(1,000,000 .\) Describe what happens to the expression as \(x\) increases.

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 3 \ln x+5 \ln y-6 \ln z $$

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 4 \ln x+7 \ln y-3 \ln z $$

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{2}(\log x+\log y) $$

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

The exponential function \(y=2^{x}\) is one-to-one and has an inverse function. Try finding the inverse function by exchanging \(x\) and \(y\) and solving for \(y .\) Describe the difficulty that you encounter in this process. What is needed to overcome this problem?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{3}\left(\log _{4} x-\log _{4} y\right) $$

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

In \(2000,\) world population was approximately 6 billion with an annual growth rate of \(1.3 \% .\) Discuss two factors that would cause this growth rate to slow down over the next ten years.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1) $$

The function \(f(x)=15,557+5259 \ln x\) models the average cost of a new car, \(f(x),\) in dollars, \(x\) years after 1989\. When was the average cost of a new car \(\$ 25,000 ?\)

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

Graph \(y=13.49(0.967)^{x}-1,\) the function for the number of O-rings expected to fail at \(x^{\circ} \mathrm{F},\) in a \([0,90,10]\) by \([0,20,5]\) viewing rectangle. If NASA engineers had used this function and its graph, is it likely they would have allowed the Challenger to be launched when the temperature was \(31^{\circ} \mathrm{F} ?\) Explain.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{5}\left(\log _{4} x-\log _{4} y\right)+2 \log _{4}(x+1) $$

The function \(f(x)=68.41+1.75 \ln x\) models the life expectancy, \(f(x),\) in years, for African-American females born \(x\) years after \(1969 .\) In which birth year was life expectancy 73.7 years? Round to the nearest year.

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

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 \(t\) 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?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{3}\left|2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right| $$

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

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.

Which one of the following is true? a. 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. b. The functions \(f(x)=3^{-x}\) and \(g(x)=-3^{x}\) have the same graph. c. \(e=2.718\) d. 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. $$\ln e$$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+\log 7+\log \left(x^{2}-1\right)-\log (x+1) $$

The \(p H\) of a solution ranges from 0 to \(14 .\) An acid solution has a pH less than 7. Pure water is neutral and has a pH of 7. Normal, unpolluted rain has a p \(H\) of about \(5.6 .\) The pH of a solution is given by $$ \mathrm{pH}=-\log x $$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve Exercises \(69-70\) An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? Express the answer as a power of \(10,\) and then round to the nearest thousandth.

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+\log 15+\log \left(x^{2}-4\right)-\log (x+2) $$

The \(p H\) of a solution ranges from 0 to \(14 .\) An acid solution has a pH less than 7. Pure water is neutral and has a pH of 7. Normal, unpolluted rain has a p \(H\) of about \(5.6 .\) The pH of a solution is given by $$ \mathrm{pH}=-\log x $$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve Exercises \(69-70\) The figure shows very acidic rain in the northeast United States. What is the hydrogen ion concentration of rainfall with a pH of \(4.2 ?\) Express the answer as a power of \(10,\) and then round to the nearest hundredthousandth.

Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

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

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{5} 13 $$

Explain how to solve an exponential equation. Use \(3^{x}=140\) in your explanation.

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

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

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{6} 17 $$

Explain how to solve a logarithmic equation. Use \(\log _{3}(x-1)=4\) in your explanation.

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

In many states, a \(17 \%\) risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do you agree with the \(17 \%\) risk as a cutoff percentage, or do you feel that the percentage should be lower or higher? Explain your answer. What blood alcohol concentration corresponds to what you believe is an appropriate percentage?

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

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{16} 57,2 $$

Evaluate each expression without using a calculator. $$e^{\ln 300}$$

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.1} 17 $$

In Exercises \(75-82,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$2^{x+1}=8$$

Use inverse properties of logarithms to simplify each expression. $$\ln e^{9 x}$$