Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $\ln \frac { x } { \sqrt [ 3 ] { x ^ { 2 } - 3 } }$
A) $\ln x + \frac { 2 } { 3 } \ln ( x - 3 )$
B) $\frac { 1 } { 3 } \ln x + \ln ( - 3 )$
C) $\frac { 1 } { 3 } \ln x - \ln ( - 3 )$
D) $\ln x - \frac { 2 } { 3 } \ln ( x - 3 )$
E) $\quad \ln x - \frac { 1 } { 3 } \ln \left( x ^ { 2 } - 3 \right)$

Question

Quizplus · Accepted Answer

We will use the properties of logarithms to rewrite the expression as a difference of logarithms:
\begin{align*}
\ln \frac { x } { \sqrt [ 3 ] { x ^ { 2 } - 3 } } &= \ln x - \ln\left(\sqrt [ 3 ] { x ^ { 2 } - 3 }ight)^{-1/3}\
&= \ln x - \frac{1}{3} \ln\left(x^2-3ight)
\end{align*}
Therefore, the answer is E.

Use the Properties of Logarithms to Expand the Expression as a Sum