## Precalculus (6th Edition) Blitzer

$\displaystyle \frac{1}{3}\ln x-\frac{1}{3}$
Basic logarithmic properties:$\left\{\begin{array}{l} \log_{b}1=0\\ \log_{b}b=1\\ \log_{b}b^{x}=x\\ b^{\log_{b}}x=x \end{array}\right.$ Rules: The Product Rule: $\log_{b}(MN)=\log_{b}\mathrm{M}+\log_{b}\mathrm{N}$ The Quotient Rule: $\displaystyle \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$ The Power Rule: $\log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$ --- $\displaystyle \ln\sqrt[3]{\frac{x}{e}}=\ln(\frac{x}{e})^{1/3}=\qquad$... apply $:$ Power Rule $=\displaystyle \frac{1}{3}\ln(\frac{x}{e})\qquad$... apply $:$ Quotient Rule $=\displaystyle \frac{1}{3}[\ln x-\ln e]\qquad$... apply $: \log_{b}b=1, (\ln e=\log_{e}e)$ $=\displaystyle \frac{1}{3}[\ln x-1]$ $=\displaystyle \frac{1}{3}\ln x-\frac{1}{3}$