Chapter 4

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to the nearest thousandth. Value \(x=9.2\) \(x=-\frac{3}{4}\) \(x=0.02\) \(x=200\) Function \(f(x)=e^{x}\)

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{3} \sqrt{x}$$.

Solve the equation for \(x.\) $$\log _{3} 3^{-5}=x$$

Solve the exponential equation. $$5\left(8^{x}\right)=325$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to the nearest thousandth. Value \(x=9.2\) \(x=-\frac{3}{4}\) \(x=0.02\) \(x=200\) Function \(f(x)=e^{-x}\)

The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is approximately 1.03323 kilograms per square centimeter, and this pressure is called one atmosphere. Variations in weatherconditions cause changes in the atmospheric pressure of up to ±5 percent. The ordered pairs \((h, p)\) give the pressures \(p\) (in atmospheres) for various altitudes \(h\) (in kilometers). (Spreadsheet at LarsonPrecalculus.com) $$(0, 1), (10, 0.25), (20, 0.06), (5, 0.55), (15, 0.12), (25, 0.02)$$ (a) Use the regression feature of a graphing utility to attempt to find the logarithmic model \(p=a+b \ln h\) for the data. Explain why the result is an error message. (b) Use the regression feature of the graphing utility to find the logarithmic model \(h=a+b \ln p\) for the data. (c) Use the graphing utility to plot the data and graph the logarithmic model in the same viewing window. (d) Use the model to estimate the altitude at which the pressure is 0.75 atmosphere. (e) Use the graph in part (c) to estimate the pressure at an altitude of 13 kilometers.

The table shows the annual sales \(S\) (in billions of dollars) of Starbucks for the years from 2009 through 2013. (Source: Starbucks Corp.) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Sales, } S \\\\\hline 2009 & 9.77 \\\\\hline 2010 & 10.71 \\\\\hline 2011 & 11.70 \\\\\hline 2012 & 13.30 \\\\\hline 2013 & 14.89 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find an exponential model for the data. Let \(t\) represent the year, with \(t=9\) corresponding to 2009. (b) Rewrite the model from part (a) as a natural exponential model. (c) Use the natural exponential model to predict the annual sales of Starbucks in \(2018 .\) Is the value reasonable?

The populations \(P\) (in thousands) of Antioch, California, from 2006 through 2012 can be modeled by \(P=90 e^{0.013 t},\) where \(t\) is the year, with \(t=6\) corresponding to \(2006 .\) (Source: U.S. Census Bureau) (a) According to the model, was the population of Antioch increasing or decreasing from 2006 through \(2012 ?\) Explain your reasoning. (b) What were the populations of Antioch in 2006 \(2009,\) and \(2012 ?\) (c) According to the model, when will the population of Antioch be approximately \(116,000 ?\)

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{5}\left(\frac{x}{2}\right)$$.

Solve the equation for \(x.\) $$\log _{8} x=\log _{8} 10^{-1}$$

Solve the exponential equation. $$2^{x+3}=256$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to the nearest thousandth. Value \(x=9.2\) \(x=-\frac{3}{4}\) \(x=0.02\) \(x=200\) Function \(g(x)=50 e^{4 x}\)

A beaker of liquid at an initial temperature of \(78^{\circ} \mathrm{C}\) is placed in a room at a constant temperature of \(21^{\circ} \mathrm{C}\) The temperature of the liquid is measured every 5 minutes for a period of \(\frac{1}{2}\) hour. The results are recorded in the table, where \(t\) is the time (in minutes) and \(T\) is the temperature (in degrees Celsius). $$\begin{array}{|c|c|}\hline \text { Time, } t 0 & \text { Temperature, } T\\\\\hline 0 & 78.0^{\circ} \\\5 & 66.0^{\circ} \\\10 & 57.5^{\circ} \\\15 & 51.2^{\circ} \\\20 & 46.3^{\circ} \\\25 & 42.5^{\circ} \\\30 & 39.6^{\circ} \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a linear model for the data. Use the graphing utility to plot the data and graph the model in the same viewing window. Do the data appear linear? Explain. (b) Use the regression feature of the graphing utility to find a quadratic model for the data. Use the graphing utility to plot the data and graph the model in the same viewing window. Do the data appear quadratic? Even though the quadratic model appears to be a good fit, explain why it might not be a good model for predicting the temperature of the liquid when \(t=60.\) (c) The graph of the temperature of the room should be an asymptote of the graph of the model. Subtract the room temperature from each of the temperatures in the table. Use the regression feature of the graphing utility to find an exponential model for the revised data. Add the room temperature to this model. Use the graphing utility to plot the original data and graph the model in the same viewing window. (d) Explain why the procedure in part (c) was necessary for finding the exponential model.

The table shows the populations (in millions) of five countries in 2013 and the projected populations (in millions) for \(2025 .\) $$\begin{array}{|c|c|c|} \hline \text { Country } & 2013 & 2025 \\ \hline \text { Australia } & 22.3 & 25.1 \\ \hline \text { Canada } & 34.6 & 37.6 \\ \hline \text { Hungary } & 9.9 & 9.6 \\ \hline \text { Philippines } & 105.7 & 128.9 \\ \hline \text { Turkey } & 80.7 & 90.5 \\ \hline \end{array}$$ (a) Find the exponential growth or decay model, \(y=a e^{b t}\) or \(y=a e^{-b t},\) for the population of each country, where \(t\) is the year, with \(t=13\) corresponding to \(2013 .\) Use the model to predict the population of each country in 2040 (b) You can see that the populations of Canada and the Philippines are growing at different rates. What constant in the equation \(y=a e^{b t}\) is determined by these different growth rates? Discuss the relationship between the different growth rates and the magnitude of the constant. (c) The population of Turkey is increasing while the population of Hungary is decreasing. What constant in the equation \(y=a e^{b t}\) reflects this difference? Explain.

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{1 / 3}\left(\frac{x}{3}\right)$$.

Solve the equation for \(x.\) $$\log _{4} 4^{3}=\log _{4} x$$

Solve the exponential equation. $$4^{x-1}=256$$

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to the nearest thousandth. Value \(x=9.2\) \(x=-\frac{3}{4}\) \(x=0.02\) \(x=200\) Function \(h(x)=-5.5 e^{-x}\)

The table shows the percents \(P\) of women in different age groups (in years) who have been married at least once. (Source: U.S. Census Bureau) $$\begin{array}{|c|c|}\hline \text { Age group } & \text { Percent, } P\\\\\hline 18-24 & 14.6 \\\25-29 & 49.0 \\\30-34 & 70.3 \\\35-39 & 79.9 \\\40-44 & 85.0 \\\45-49 & 87.0 \\\50-54 & 89.5 \\\55-59 & 91.1 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a logistic model for the data. Let \(x\) represent the midpoint of the age group. (b) Use the graphing utility to graph the model with the original data. How closely does the model represent the data?

The populations \(P\) (in thousands) of Cameron County, Texas, from 2006 through 2012 can be modeled by \(P=339.2 e^{k t}\) where \(t\) is the year, with \(t=6\) corresponding to 2006 In \(2011,\) the population was \(412,600 .\) (Source: U.S. Census Bureau) (a) Find the value of \(k\) for the model. Round your result to four decimal places. (b) Use your model to predict the population in 2018 .

Use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\log _{4} 8$$.

Use the properties of logarithms to simplify the expression. $$\log _{4} 4^{3 x}$$

Solve the logarithmic equation. $$\ln x-\ln 5=0$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=\left(\frac{5}{2}\right)^{x}$$

The table shows the lengths \(y\) (in centimeters) of yellowtail snappers caught off the coast of Brazil for different ages (in years). (Source: Brazilian Journal of Oceanography) $$\begin{array}{|c|c|}\hline \text { Age, }x & \text { Length, } y\\\\\hline 1 & 11.21 \\\2 & 20.77 \\\3 & 28.94 \\\4 & 35.92 \\\5 & 41.87 \\\6 & 46.96 \\\7 & 51.30 \\\8 & 55.01 \\\9 & 58.17 \\\10 & 60.87 \\\11 & 63.18 \\\12 & 65.15 \\\13 & 66.84 \\\14 & 68.27 \\\15 & 69.50 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find a logistic model and a power model for the data. (b) Use the graphing utility to graph each model from part (a) with the data. Use the graphs to determine which model better fits the data. (c) Use the model from part (b) to predict the length of a 17-year-old yellowtail snapper.

The populations \(P\) (in thousands) of Pineville, North Carolina, from 2006 through 2012 can be modeled by \(P=5.4 e^{k t},\) where \(t\) is the year, with \(t=6\) corresponding to \(2006 .\) In \(2008,\) the population was 7000. (Source: U.S. Census Bureau) (a) Find the value of \(k\) for the model. Round your result to four decimal places. (b) Use your model to predict the population in 2018 .

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\log _{9} 243$$.

Use the properties of logarithms to simplify the expression. $$6^{\log _{6} 36 x}$$

Solve the logarithmic equation. $$\ln x-\ln 2=0$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=\left(\frac{5}{2}\right)^{-x}$$

Determine whether the statement is true or false. Justify your answer. The exponential model \(y=a e^{b x}\) represents a growth model when \(b>0.\)

Carbon 14 ( \(^{14} \mathrm{C}\) ) dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of \(^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \(^{14} \mathrm{C}\) absorbed by a tree growing today. \(\mathrm{A}\) piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal, given that the half-life of \(^{14} \mathrm{C}\) is about 5700 years?

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\log _{2} 4^{2} \cdot 3^{4}$$.

Use the properties of logarithms to simplify the expression. $$3 \log _{2} \frac{1}{2}$$

Solve the logarithmic equation. $$\ln x=-9$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=6^{x}$$

The half-life of radioactive radium \(\left(^{226} \mathrm{Ra}\right)\) is 1600 years. What percent of a present amount of radioactive radium will remain after 100 years?

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\log _{3} 9^{2} \cdot 2^{4}$$.

Use the properties of logarithms to simplify the expression. $$\frac{1}{4} \log _{4} 16$$

Solve the logarithmic equation. $$\ln x=-14$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2^{x-1}$$

In your own words, explain how to fit a model to a set of data using a graphing utility.

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\ln 5 e^{6}$$.

Sketch the graph of \(f .\) Then use the graph of \(f\) to sketch the graph of \(g.\) $$\begin{aligned}&f(x)=6^{x}\\\&g(x)=\log _{6} x\end{aligned}$$

Solve the logarithmic equation. $$\log _{x} 625=4$$

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=3^{x+2}$$

A new laptop computer that sold for \(\$ 1200\) in 2014 has a book value \(V\) of \(\$ 650\) after 2 years. (a) Find a linear model for the value \(V\) of the laptop. (b) Find an exponential model for the value \(V\) of the laptop. Round the numbers in the model to four decimal places. (c) Use a graphing utility to graph the two models in the same viewing window. (d) Which model represents a greater depreciation rate in the first year? (e) For what years is the value of the laptop greater using the linear model? the exponential model?

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\ln 8 e^{3}$$.

Sketch the graph of \(f .\) Then use the graph of \(f\) to sketch the graph of \(g.\) $$\begin{aligned}&f(x)=5^{x}\\\&g(x)=\log _{5} x\end{aligned}$$

Solve the logarithmic equation. $$\log _{x} 81=2$$