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Chapter 3: Problem 35
Use the properties of logarithms to simplify the expression. $$\log _{\pi} \pi$$
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Let \(f(x)=\log _{a} x\) and \(g(x)=a^{x},\) where \(a>1\) (a) Let \(a=1.2\) and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of \(a\) for which the two graphs have one point of intersection. (c) Determine the value(s) of \(a\) for which the two graphs have two points of intersection.
Use the following information for determining sound intensity. The level of sound \(\boldsymbol{\beta}\), in decibels, with an intensity of \(I\), is given by \(\boldsymbol{\beta}=10 \log \left(I / I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 65 and 66 , find the level of sound \(\boldsymbol{\beta}\). (a) \(I=10^{-11}\) watt per \(\mathrm{m}^{2}\) (rustle of leaves) (b) \(I=10^{2}\) watt per \(\mathrm{m}^{2}\) (jet at 30 meters) (c) \(I=10^{-4}\) watt per \(\mathrm{m}^{2}\) (door slamming) (d) \(I=10^{-2}\) watt per \(\mathrm{m}^{2}\) (siren at 30 meters)
he value \(V\) (in millions of dollars) of a famous painting can be modeled by \(V=10 e^{k t},\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . In 2008 , the same painting was sold for \(\$ 65\) million. Find the value of \(k,\) and use this value to predict the value of the painting in 2014 .
Use the Richter scale \(R=\log \frac{l}{I_{0}}\) for measuring the magnitudes of earthquakes. Find the intensity \(I\) of an earthquake measuring \(R\) on the Richter scale (let \(I_{0}=1\) ). (a) Southern Sumatra, Indonesia in \(2007, R=8.5\) (b) Illinois in \(2008, R=5.4\) (c) Costa Rica in \(2009, R=6.1\)
Complete the table for the time \(t\) (in years) necessary for \(P\) dollars to triple if interest is compounded continuously at rate \(r\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline r & 2 \% & 4 \% & 6 \% & 8 \% & 10 \% & 12 \% \\ \hline t & & & & & & \\ \hline \end{array} $$
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