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Chapter 3: Problem 17
Solve the exponential equation algebraically. Approximate the result to three decimal places. $$e^{x}=e^{x^{2}-2}$$
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Solve the equation algebraically. Round your result to three decimal places. Verify your answer using a graphing utility. $$e^{-2 x}-2 x e^{-2 x}=0$$
Sound Intensity The relationship between the number of decibels \(\beta\) and the intensity of a sound \(I\) in watts per square meter is \(\beta=10 \log \left(\frac{I}{10^{-12}}\right)\) (a) Determine the number of decibels of a sound with an intensity of 1 watt per square meter. (b) Determine the number of decibels of a sound with an intensity of \(10^{-2}\) watt per square meter. (c) The intensity of the sound in part (a) is 100 times as great as that in part (b). Is the number of decibels 100 times as great? Explain.
The values \(y\) (in billions of dollars) of U.S. currency in circulation in the years 2000 through 2010 can be modeled by \(y=-611+507\) ln \(t, 10 \leq t \leq 20\) where \(t\) represents the year, with \(t=10\) corresponding to 2000. During which year did the value of U.S. currency in circulation exceed \(\$ 690\) billion? (Source: Board of Governors of the Federal Reserve System )
True or False? In Exercises 83 and \(84,\) determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{6} x\) is a reflection of the graph of \(g(x)=6^{x}\) in the \(x\) -axis.
In Exercises \(97-102,\) determine whether the statement is true or false given that \(f(x)=\ln x .\) Justify your answer. If \(f(x) < 0,\) then \(0 < x < 1\)
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