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Chapter 3: Problem 78
Condense the expression to the logarithm of a single quantity. $$3 \log _{3} x+4 \log _{3} y-4 \log _{3} z$$
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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. The \(\mathrm{pH}\) of a solution is decreased by one unit. The hydrogen ion concentration is increased by what factor?
The total interest \(u\) paid on a home mortgage of \(P\) dollars at interest rate \(r\) for \(t\) years is \(u=P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right]\) Consider a $$\$ 120,000$$ home mortgage at \(7 \frac{1}{2} \%\). (a) Use a graphing utility to graph the total interest function. (b) Approximate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paying twice as much in interest charges as the size of the mortgage?
A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the pack will be modeled by the logistic curve \(p(t)=\frac{1000}{1+9 e^{-0.1656 t}}\) where \(t\) is measured in months (see figure). (a) Estimate the population after 5 months. (b) After how many months will the population be \(500 ?\) (c) Use a graphing utility to graph the function. Use the graph to determine the horizontal asymptotes, and interpret the meaning of the asymptotes in the context of the problem.
The sales \(S\) (in thousands of units) of a new CD burner after it has been on the market for \(t\) years are modeled by \(S(t)=100\left(1-e^{k t}\right) .\) Fifteen thousand units of the new product were sold the first year. (a) Complete the model by solving for \(k\). (b) Sketch the graph of the model. (c) Use the model to estimate the number of units sold after 5 years.
Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x \ln \left(\frac{1}{x}\right)-x=0$$
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