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Chapter 3: Problem 40
Find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.) $$\ln \sqrt[4]{e^{3}}$$
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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.
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{3} x+\log _{3}(x-8)=2$$
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log 8 x-\log (1+\sqrt{x})=2$$
Use the Richter scale \(R=\log \frac{l}{I_{0}}\) for measuring the magnitudes of earthquakes. Find the magnitude \(R\) of each earthquake of intensity \(I\) (let \(I_{0}=1\) ). (a) \(I=199,500,000\) (b) \(I=48,275,000\) (c) \(I=17,000\)
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{2} x+\log _{2}(x+2)=\log _{2}(x+6)$$
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