Chapter 5

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. entering a room

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log x+\log (x-21)=2$$

Suppose that the concentration of a bacterial sample is \(50,000\) bacteria per milliliter. If the concentration triples in 4 days, how long will it take for the concentration to reach \(85,000\) bacteria per milliliter?

Solve each equation. $$(\sqrt[3]{5})^{-x}=\left(\frac{1}{5}\right)^{x+2}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 1247$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. climbing the stairs

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log x+\log (3 x-13)=1$$

Suppose that the cost of photovoltaic cells each year after 1980 was \(75 \%\) as much as the year prior. If the cost was \(\$ 30 /\) watt in \(1980,\) model their price in dollars with an exponential function, where \(x\) corresponds to years after \(1980 .\) Then estimate the year when the price of photovoltaic cells was \(\$ 1.00\) per watt.

Solve each equation. $$(\sqrt{2})^{-2 x}=\left(\frac{1}{2}\right)^{2 x+3}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 0.783$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. wrapping a package

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (4 x-2)-\ln 4=-\ln (x-2)$$

Suppose that when a ball is dropped, the height of its first rebound is about \(80 \%\) of the initial height that it was dropped from, the second rebound is about \(80 \%\) as high as the first rebound, and so on. If this ball is dropped from 12 feet in the air, model the height in feet of each rebound with an exponential function \(H(x),\) where \(x=0\) represents the initial height, \(x=1\) represents the height on the first rebound, and so on. Find the height of the third rebound. Determine which rebound had a height of about 2.5 feet.

Solve each equation. $$(\sqrt[4]{3})^{-x}=\left(\frac{1}{3}\right)^{x-1}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 0.014$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. putting on a coat

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (5+4 x)-\ln (3+x)-\ln 3=0$$

In 1666 the village of Eyam, located in England, experienced an outbreak of the Great Plague. Out of 261 people in the community, only 83 survived. The table shows a function \(f\) that computes the number of people who had not (yet) been infected after \(x\) days. $$\begin{array}{|c|r|r|r|r|}\hline x & 0 & 15 & 30 & 45 \\\\\hline f(x) & 254 & 240 & 204 & 150 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 60 & 75 & 90 & 125 \\\\\hline f(x) & 125 & 103 & 97 & 83\\\\\hline\end{array}$$ (a) Use a table to represent a function \(g\) that computes the number of people in Eyam who were infected after \(x\) days. (b) Write an equation that shows the relationship between \(f(x)\) and \(g(x)\) (c) Use graphing to decide which equation represents \(g(x)\) better \(y_{1}=\frac{171}{1+18.6 e^{-0.0747 x}}\) or \(y_{2}=18.3(1.024)^{x}\) (d) Use your results from parts (b) and (c) to find a formula for \(f(x)\)

Solve each equation. $$6^{1-x}=\left(\frac{1}{36}\right)^{2 x}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 28^{3}$$

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=3 x-4$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{5}(x+2)+\log _{5}(x-2)=1$$

Evaluate each logarithm in three ways: (a) Use the definition of logarithm in Section 5.3 to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on your calculator. (c) Support the result of part (a) by locating the appropriate point on the graph of the function y=\log _{a} x. $$\log _{16}\left(\frac{1}{8}\right)$$

The table lists heart disease death rates per \(100,000\) people for selected ages. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Age } & 30 & 40 & 50 & 60 & 70 \\\ \hline \text { Death Rate } & 8.0 & 29.6 & 92.9 & 246.9 & 635.1\\\\\hline \end{array}$$ (a) Make a scatter diagram of the data in the window \([25,75]\) by \([-100,700]\). (b) Find an exponential function \(f\) that models the data. (c) Estimate the heart disease death rate for people who are 80 years old.

Solve each equation. $$\left(\frac{3}{5}\right)^{-x}=\left(\frac{9}{25}\right)^{1-5 x}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln (47 \times 93)$$

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=4 x-5$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{2}(x-7)+\log _{2} x=3$$

The following table shows the revenue in billions of dollars for iTunes in various years. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 2008 & 2009 & 2010 & 2011 & 2012 \\ \hline \begin{array}{l} \text { Revenue } \\ \text { (\$billions) } \end{array} & 6.0 & 7.0 & 9.5 & 11.8 & 15.7 \\\\\hline\end{array}$$ (a) Use exponential regression to approximate constants \(C\) and \(a\) so that \(f(x)=C a^{x-2008}\) models the data, where \(x\) is the year. (b) Support your answer by graphing \(f\) and the data.

For each exponential function f, find f^{-1} analytically and graph both f and f^{-1} in the same viewing window. $$f(x)=4^{x}-3$$

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(2^{x+1}=8\) (b) \(2^{x+1}>8\) (c) \(2^{x+1}<8\)

Find the pH for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Grapefruit, \(6.3 \times 10^{-4}\)

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=x^{3}+1$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{7}(4 x)-\log _{7}(x+3)=\log _{7} x$$

The following table shows the average Valentine's Day spending in dollars per consumer for various years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2010 & 2011 & 2012 & 2013 \\\ \hline \text { Spending ( } \$ \text { ) } & 103 & 116 & 126 & 131 \\\\\hline\end{array}$$ (a) Use exponential regression to approximate values for \(a\) and \(b\) so that \(f(x)=a+b \ln x\) models the data, where \(x=1\) corresponds to \(2010, x=2\) to 2011 and so on. (b) Use your function to estimate average spending in 2012 and compare to the value in the table.

For each exponential function f, find f^{-1} analytically and graph both f and f^{-1} in the same viewing window. $$f(x)=\left(\frac{1}{2}\right)^{x}-5$$

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(3^{2-x}=9\) (b) \(3^{2-x}>9\) (c) \(3^{2-x}<9\)

Find the pH for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Limes, \(1.6 \times 10^{-2}\)

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=-x^{3}-2$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{2}(2 x)+\log _{2}(x+2)=\log _{2} 16$$

In real life, populations of bacteria, insects, and animals do not continue to grow indefinitely. Initially, population growth may be slow. Then, as their numbers increase, so does the rate of growth. After a region has become heavily populated or saturated, the population usually levels off because of limited resources. This type of growth may be modeled by a logistic function represented by $$f(x)=\frac{c}{1+a e^{-b x}}$$ where \(a, b,\) and \(c\) are positive constants. As age increases, so does the likelihood of coronary heart disease (CHD). The fraction of people \(x\) years old with some CHD is approximated by $$f(x)=\frac{0.9}{1+271 e^{-0.122 x}}$$ (Source: Hosmer, D. and S. Lemeshow, Applied Logistic Regression, John Wiley and Sons.) (a) Evaluate \(f(25)\) and \(f(65) .\) Interpret the results. (b) At what age does this likelihood equal \(50 \% ?\)

For each exponential function f, find f^{-1} analytically and graph both f and f^{-1} in the same viewing window. $$f(x)=-10^{x}+4$$

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(27^{4 x}=9^{x+1}\) (b) \(27^{4 x}>9^{x+1}\) (c) \(27^{4 x}<9^{x+1}\)

Find the pH for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Crackers, \(3.9 \times 10^{-9}\)

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=x^{2}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln e^{x}-2 \ln e=\ln e^{4}$$

Tree Growth The height of a tree in feet after \(x\) years is modeled by $$f(x)=\frac{50}{1+47.5 e^{-0.22 x}}$$ (a) Make a table for \(f\) starting at \(x=10\) and incrementing by \(10 .\) What seems to be the maximum height? (b) Graph \(f\) and identify the horizontal asymptote. Explain its significance. (c) After how long was the tree 30 feet tall?

For each exponential function f, find f^{-1} analytically and graph both f and f^{-1} in the same viewing window. $$f(x)=-e^{x}+6$$

Find the pH for each substance with the given hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration. Sodium hydroxide (lye), \(3.2 \times 10^{-14}\)

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(32^{x}=16^{1-x}\) (b) \(32^{x}>16^{1-x}\) (c) \(32^{x}<16^{1-x}\)