91Ó°ÊÓ

Solve each exponential equation . Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7^{2 x+1}=3^{x+2} $$

Exercises \(47-52\) present data in the form of tables. For each data set shown by the table, a. Create a scatter plot for the data. b. Use the scatter plot to determine whether an exponential function, a logarithmic function, or a linear function is the best choice for modeling the data. (If applicable, in Exercise \(72,\) you will use your graphing utility to obtain these functions.) Hamachiphobia $$ \begin{array}{ccc} {} & {\text { Percentage }} & {\text { Percentage } W h o} \\ {} & {\text { Who Won't }} & {\text { Don't Approve of }} \\ {\text { Generation }} & {\text { Try Sushi }} & {\text { Marriage Equality }} \\\ {\text { Millennials }} & {42} & {36} \\ {\text { Gen } X} & {52} & {36} \\ {\text { Boomers }} & {60} & {49} \\ {\text { Silent/Greatest }} & {72} & {66} \\ {\text { Generation }} \end{array} $$

Explaining the Concepts How can you tell whether an exponential model describes exponential growth or exponential decay?

Explaining the Concepts Suppose that a population that is growing exponentially increases from \(800,000\) people in 2010 to \(1,000,000\) people in \(2013 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.1} 17 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because carbon- 14 decays exponentially, carbon dating can determine the ages of ancient fossils.

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In \(2006,\) Canada's population exceeded Uganda's by 4.9 million.

Exercises \(86-88\) will help you prepare for the material covered in the first section of the next chapter. $$ \text { Simplify: } \frac{17 \pi}{6}-2 \pi $$

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

In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ \ln e^{7} $$