Chapter 4

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(s)=2 s-\frac{9}{5}$$

The following table gives the sales, in billions of current dollars, for restaurants in the United States for selected years from 1970 to \(2005 .\) (Source: National Restaurant Association Fact Sheet, 2005 ) $$\begin{aligned}&\text { Year } 1970 \quad 1985 \quad 1995 \quad 2005\\\&\text { Sales } 42.8 \quad 173.7 \quad 295.7 \quad 475.8\end{aligned}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(f(x)=C a^{x}\) that best fits the data. Let \(x\) be the number of years since 1970 (b) Why must \(a\) be greater than 1 in your model? (c) Using your model, what are the projected sales for restaurants in the year \(2008 ?\) (d) Do you think this model will be accurate over the long term? Explain.

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

Sketch the graph of each function and find (a) the \(y\) -intercept; (b) the domain and range; (c) the horizontal asymptote;and (d) the behavior of the function as \(x\) approaches\(\pm \infty .\) $$f(x)=2^{-x}$$

Evaluate the expression to four decimal places using a calculator. $$\log 1400$$

Tourism The following table shows the tourism revenue for China, in billions of dollars, for selected years since \(1990 .\) (Source: World Tourism Organization)$$\begin{array}{|c|c|}\hline \text { Year } & \begin{array}{c}\text { Revenue } \\\\\text { (billions of dollars) }\end{array} \\\\\hline 1990 & 2.218 \\\1995 & 8.733 \\\1996 & 10.200 \\\1998 & 12.602 \\\2000 & 16.231 \\\2002 & 20.385\\\ \hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(f(x)=C a^{x}\) that best fits the data. Let \(x\) be the number of years since \(1990 .\) (b) Using this model, what is the projected revenue from tourism in the year \(2008 ?\) (c) Do you think this model will be accurate over the long term? Explain.

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=x^{3}-6$$

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

Sketch the graph of each function and find (a) the \(y\) -intercept; (b) the domain and range; (c) the horizontal asymptote;and (d) the behavior of the function as \(x\) approaches\(\pm \infty .\) $$f(x)=3-2^{x}$$

Evaluate the expression to four decimal places using a calculator. $$\log 2500$$

The spread of a disease can be modcled by a logistic function. For example, in carly 2003 there was an outbreak of an illness called SARS (Severe Acute Respiratory Syndrome) in many parts of the world. The following table gives the total momber of cases in Canada for the wecks following March 20,2003 (Source: World Health Organization) (Note: The total number of cases dropped from 149 to 140 between weeks 3 and 4 because some of the cases thought to be SARS were reclassified as other discases.) $$\begin{array}{|c|c|}\hline\text { Weeks since } & \\\\\text { March } 20,2003 & \text { Total Cases } \\\0 & 9 \\\1 & 62 \\\2 & 132 \\\3 & 149 \\\4 & 140 \\\5 & 216 \\\6 & 245 \\\7 & 252 \\\8 & 250 \\\\\hline\end{array}$$ (a) Explain why a logistic function would suit this data well. (b) Make a scatter plot of the data and find the logistic function of the form \(f(x)=\frac{\epsilon}{1+a \varepsilon^{-1}}\) that best fits the data. (c) What docs \(c\) signify in your model? (d) The World Health Organization declared in July 2003 that SARS no longer posed a threat in Canada. By analyring this data, explain why that would be so.

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-x^{3}+4$$

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

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\frac{1}{3} \log _{4} 8 x^{9}-\log _{4} x^{2}$$

Evaluate the expression to four decimal places using a calculator. $$2 \log \frac{1}{5}$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{1}{2} x-4$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{3}(x+4)=2$$

Sketch the graph of each function and find (a) the \(y\) -intercept; (b) the domain and range; (c) the horizontal asymptote;and (d) the behavior of the function as \(x\) approaches\(\pm \infty .\) $$f(x)=7 e^{x}$$

Evaluate the expression to four decimal places using a calculator. $$-\ln \frac{2}{3}$$

The following table gives the temperature, in degrees Celsius, of a cup of hot water sitting in a room with constant temperature. The data was collected over a period of 30 minutes. (Source: www.phys. unt.edu, Dr. James A. Roberts)$$\begin{array}{|c|c|} \hline\text { Time } & \text { Temperature } \\\\(\mathrm{min}) & (\text { degrees Celsius }) \\ \hline0 & 95 \\\1 & 90.4 \\\5 & 84.6 \\\10 & 73 \\\15 & 64.7 \\\20 & 59 \\\25 & 54.5 \\\29 & 51.4\\\\\hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(f(t)=C a^{2}\) that best fits the data. Let \(t\) be the number of minutes the water has been cooling. (b) Using your modicl, what is the projected temperature of the water after 1 hour?

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-\frac{3}{4} x+2$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{5}(x+3)=1$$

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\ln \left(x^{2}-9\right)-\ln (x+3)$$

Sketch the graph of each function and find (a) the \(y\) -intercept; (b) the domain and range; (c) the horizontal asymptote;and (d) the behavior of the function as \(x\) approaches\(\pm \infty .\) $$g(x)=-4 e^{2 x}$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{3} 1.25$$

The following table gives the price per barrel of crude oil for selected years from 1992 to 2006 (Source: www.ioga.com/special/crudeoil-Hist.htm) $$\begin{array}{|c|c|}\hline\text { Year } & \begin{array}{c}\text { Price } \\\\\text { (dollars) }\end{array} \\\\\hline 1992 & 19.25 \\\1996 & 20.46 \\\2000 & 27.40 \\\2004 & 37.41 \\\2006 & 58.30\\\ \hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(P(t)=C a^{t}\) that best fits the data. Let \(t\) be the number of years since 1992 (b) Using your model, what is the projected price per barrel of crude oil in \(2009 ?\)

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=-x^{2}+8, x \geq 0$$

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

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\ln \left(x^{2}-1\right)-\ln (x-1)$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{3} 2.75$$

The following table gives the total amount spent by all candidates in each presidential election, beginning in \(1988 .\) Each amount listed is in millions. (Source: Federal Election Commission) $$\begin{array}{|c|c|} \hline\text { Year } & \text { Price } \\\\\hline1988 & 495 \\\1988 & 550 \\\1992 & 560 \\\1996 & 649.5 \\\2000 & 1,016.5 \\\2004 & 1,016.5 \\\ \hline\end{array}$$ (a) Make a scatter plot of the data and find the exponential function of the form \(P(t)=C a^{2}\) that best fits the data. Let \(t\) be the number of years since 1988 (b) Using your model, what is the projected total amount all candidates will spend during the 2012 presidential election?

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=-x^{2}+3, x \leq 0$$

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

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\frac{1}{2}\left[\log \left(x^{2}-1\right)-\log (x+1)\right]+\log x$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{5} 0.5$$

The cost of removing chemicals from drinking water depends on how much of the chemical can safcly be left behind in the water. The following table lists the annual removal costs for arsenic in terms of the concentration of arsenic in the drinking water. (Source: Environmental Protection Agency) $$\begin{array}{|c|c|}\hline\text { Arsenic Concentration } & \text { Annual Cost } \\\\\text { (micrograms per liter) } & \text { (millions of dollars) } \\\\\hline 3 & 645 \\\5 & 379 \\\10 & 166 \\\20 & 65\\\ \hline\end{array}$$ (a) Interpret the data in the table. What is the relation between the amount of arsenic left behind in the removal process and the annual cost? (One microgram is equal to \(10^{-6}\) gram.) (b) Make a scatter plot of the data and find the exponential function of the form \(C(x)=C a^{*}\) that best fits the data. Here, \(x\) is the arscnic concentration. (c) Why must \(a\) be less than 1 in your model? (d) Using your model, what is the annual cost to obtain an arsenic concentration of 12 micrograms per liter? (e) It would be best to have the smallest possible amount of arsenic in the drinking water, but the cost may be prohibitive. Use your model to calculate the annual cost of processing such that the concentration of arsenic is only 2 micrograms per liter of water. Interpret your result.

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=x^{2}-5, x \leq 0$$

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

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{5} 0.65$$

The following data gives the percentage of women who smoked during pregnancy for selected years from 1994 to \(2002 .\) (Sournce: National Center for Health Statistics) $$\begin{array}{|c|c|} \hline\text { Year } & \text { Percent Smoking } \\\\\text { Yuring Pregnancy } \\ \hline1994 & 14.6 \\\1996 & 13.6 \\\1998 & 12.9 \\\2000 & 12.2 \\\2001 & 12.0 \\\2002 & 11.4\\\\\hline\end{array}$$ (a) From examining the table, what is the general relationship between the year and the percentage of women smoking during pregnancy? (b) Let \(t\) be the number of years after \(1993 .\) Here, \(t\) starts at 1 because in 0 is undefined. Make a scatter plot of the data and find the natural logarithmic function of the form \(p(t)=a \ln t+b\) that best fits the data. Why must a be negative? (c) Project the percentage of women who will smoke during pregnancy in the year 2007.

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=x^{2}-6, x \geq 0$$

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

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\frac{3}{2} \log 16 x^{4}-\frac{1}{2} \log y^{8}$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{2} 12$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-2 x^{3}+7$$

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

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{2} 20$$

The value \(c\) in the logistic function \(f(x)=\frac{\epsilon}{1+a c^{-2}}\) is sometimes called the carrying capacity. Can you give a reason why this term is used?

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=3 x^{3}-5$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log _{5} x=1-\log _{5}(x-4)$$