Answer
$\displaystyle \frac{1}{3}\ln x-\frac{1}{3}$
Work Step by Step
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: $\log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
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$\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] $
$=\displaystyle \frac{1}{3}\ln x-\frac{1}{3}$
