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Chapter 4: Problem 8
Write each equation in its equivalent exponential form. $$\log _{5} 125=y$$
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Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$\log (x+3)+\log x=1$$
Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\)
Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$3^{x}=2 x+3$$
Use the exponential growth model, \(A=A_{0} e^{k_{i}},\) to show that the time it takes a population to triple (to grow from \(A_{0}\) to \(\left.3 A_{0}\right)\) is given by \(t=\frac{\ln 3}{k}\)
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 \(2006 .\) 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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