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Chapter 4: Problem 87
Evaluate or simplify each expression without using a calculator. $$\ln 1$$
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Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\)
Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12$$ a. What was the average score on the original exam? b. What was the average score after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.
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.
Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$
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. $$2^{x+1}=8$$
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