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Chapter 3: Problem 36
Use the properties of logarithms to simplify the expression. $$9^{\log _{9} 15}$$
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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) \(I=10^{-10}\) watt per \(\mathrm{m}^{2}\) (quiet room) (b) \(I=10^{-5}\) watt per \(\mathrm{m}^{2}\) (busy street corner) (c) \(I=10^{-8}\) watt per \(\mathrm{m}^{2}\) (quiet radio) (d) \(I=10^{0}\) watt per \(\mathrm{m}^{2}\) (threshold of pain)
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x+\ln (x-2)=1$$
A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers \(y\) of cell sites from 1985 through 2008 can be modeled by \(y=\frac{237,101}{1+1950 e^{-0.355 t}}\) where \(t\) represents the year, with \(t=5\) corresponding to \(1985 .\) (Source: CTIA-The Wireless Association) (a) Use the model to find the numbers of cell sites in the years 1985,2000 , and 2006 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the number of cell sites will reach 235,000 . (d) Confirm your answer to part (c) algebraically.
Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x \ln x+x=0$$
Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Find the \(\mathrm{pH}\) if \(\left[\mathrm{H}^{+}\right]=2.3 \times 10^{-5}\).
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