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Chapter 3: Problem 89
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$3 \ln 5 x=10$$
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Determine the principal \(P\) that must be invested at rate \(r\), compounded monthly, so that $$\$ 500,000$$ will be available for retirement in \(t\) years. $$r=5 \%, t=10$$
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 \(\boldsymbol{\beta}\). Due to the installation of a muffler, the noise level of an engine was reduced from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of the muffler.
$$\$ 2500$$ is invested in an account at interest rate \(r\), compounded continuously. Find the time required for the amount to (a) double and (b) triple. $$r=0.0375$$
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.
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.
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