Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
$$\ln \frac { x } { \sqrt [ 3 ] { x ^ { 5 } - 2 } }$$
A) $\ln x + \frac { 5 } { 3 } \ln ( x - 2 )$
B) $- \frac { 2 } { 3 } \ln x + \ln ( - 2 )$
C) $- \frac { 2 } { 3 } \ln x - \ln ( - 2 )$
D) $\ln x - \frac { 5 } { 3 } \ln ( x - 2 )$
E) $\quad \ln x - \frac { 1 } { 3 } \ln \left( x ^ { 5 } - 2 \right)$

Question

Quizplus · Accepted Answer

Using the properties of logarithms, we can rewrite the expression as:

\begin{align*}
\ln \frac{x}{\sqrt[3]{x^5-2}} &= \ln x - \ln\sqrt[3]{x^5-2} \
&= \ln x - \frac13\ln(x^5-2)
\end{align*}

Therefore, the answer is E.

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