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Chapter 3: Problem 92
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln \sqrt{x-8}=5$$
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Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x+\ln (x-2)=1$$
The populations \(P\) (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by \(P=346.8 e^{k t},\) where \(t\) represents the year, with \(t=0\) corresponding to \(2000 .\) In \(2005,\) the population of Reno was about 395,000 . (Source: U.S. Census Bureau) (a) Find the value of \(k\). Is the population increasing or decreasing? Explain. (b) Use the model to find the populations of Reno in 2010 and 2015 . Are the results reasonable? Explain. (c) According to the model, during what year will the population reach \(500,000 ?\)
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 \(\left[\mathrm{H}^{+}\right]=1.13 \times 10^{-5}\).
Determine whether the statement is true or false. Justify your answer. A logistic growth function will always have an \(x\) -intercept.
The yield \(V\) (in millions of cubic feet per acre) for a forest at age \(t\) years is given by \(V=6.7 e^{-48.1 / t}\) (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 1.3 million cubic feet.
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