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Chapter 4: Problem 78
What is the natural exponential function?
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Exercises \(153-155\) will help you prepare for the material covered in the next section. a. Simplify: \(e^{\ln 3}\). b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)
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. By \(2009,\) the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.
Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)
Determine whether each statement makes sense or does not make sense, and explain your reasoning. I expanded log \(_{4} \sqrt{\frac{x}{y}}\) by writing the radical using a rational exponent and then applying the quotient rule, obtaining \(\frac{1}{2} \log _{4} x-\log _{4} y\)
By 2019 , nearly \(\$$ I out of every \)\$ 5\( spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product \)(G D P)\( going toward health care from 2007 through \)2014,\( with a projection for 2019 The data can be modeled by the function \)f(x)=1.2 \ln x+15.7\( where \)f(x)\( is the percentage of the U.S. gross domestic product going toward health care \)x\( years after \)2006 .\( Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \)2009 .\( Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \)18.5 \%$ of the U.S. gross domestic product go toward health care? Round to the nearest year.
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