91Ó°ÊÓ

For the following exercises, graph the transformation of \(f(x)=2^{x}\). Give the horizontal asymptote, the domain, and the range. $$ h(x)=2^{x}+3 $$

Prove that \(b^{x}=e^{x \ln (b)}\) for positive \(b \neq 1\)

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & -3.25 & 2 & 7.25 & 12.5 \\ \hline \end{array} $$

For the following exercises, refer to Table 7. $$\begin{array}{ccccccc}{x} & {1} & {2} & {3} & {4} & {5} & {6} \\ {f(x)} & {1125} & {1495} & {2310} & {3294} & {4650} & {6361}\end{array}$$ Write the exponential function as an exponential equation with base \(e .\)

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about \(30 \%\) each hour. To the nearest hour, what is the half-life of the drug?

Refer to Table 7. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array}$$ Write the exponential function as an exponential equation with base \(e\).

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\log _{5}(x)=2$$

For the following exercises, use logarithms to solve. $$ 3 e^{3-3 x}+6=-31 $$

For the following exercises, graph the transformation of \(f(x)=2^{x}\). Give the horizontal asymptote, the domain, and the range. $$ f(x)=2^{x-2} $$

For the following exercises, suppose \(\log _{5}(6)=a\) and \(\log _{5}(11)=b\). Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of \(a\) and \(b\). Show the steps for solving. $$ \log _{6}(55) $$