91Ó°ÊÓ

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

What does the \(y\)-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

For the following exercises, refer to Table 4.29. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} \\ \hline f(x) & {7.5} & {6} & {5.2} & {4.3} & {3.9} & {3.1} & {2.9} \\ \hline\end{array}$$ Use the logarithmic function to find the value of the function when \(x=10\).

Rewrite \(\ln (z)-\ln (x)-\ln (y)\) in compact form.

Rewrite log \(_{3}(12.75)\) to base \(e\).

Rewrite \(5^{12 x-17}=125\) as a logarithm. Then apply the change of bange of base formula to solve for \(x\) using the common log. Round to the nearest thousandth.

Solve \(216^{3 x} \cdot 216^{x}=36^{3 x+2}\) by rewriting each side with a common base.

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline x & {0.15} & {0.25} & {0.5} & {0.75} & {1} & {1.5} & {2} & {2.25} & {2.75} & {3} & { 3.5} \\ \hline f(x) & {36.21} & {28.88} & {24.39} & {18.28} & {16.5} & {12.99} & {9.91} & {8.57} & {7.23} & {5.99} & {4.81} \\ \hline\end{array}$$

Evaluate \(\log (10,000,000)\) without using a calculator.

Evaluate \(\ln (0.716)\) using a calculator. Round to the nearest thousandth.