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Chapter 3: Problem 97
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$6 \log _{3}(0.5 x)=11$$
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Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$
Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x-\ln (x+1)=2$$
Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$3-\ln x=0$$
The numbers \(y\) of freestanding ambulatory care surgery centers in the United States from 2000 through 2007 can be modeled by \(y=2875+\frac{2635.11}{1+14.215 e^{-0.8038 t}}, \quad 0 \leq t \leq 7\) where \(t\) represents the year, with \(t=0\) corresponding to 2000 . (Source: Verispan) (a) Use a graphing utility to graph the model. (b) Use the trace feature of the graphing utility to estimate the year in which the number of surgery centers exceeded \(3600 .\)
In a group project in learning theory, a mathematical model for the proportion \(P\) of correct responses after \(n\) trials was found to be \(P=0.83 /\left(1+e^{-0.2 n}\right)\) (a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. (c) After how many trials will \(60 \%\) of the responses be correct?
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