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

Using the properties of logarithms, the given expression $\ln \frac { x } { \sqrt [ 3 ] { x ^ { 2 } - 3 } }$ can be expanded as $\ln x - \ln \left( \sqrt [ 3 ] { x ^ { 2 } - 3 } \right)$. Applying the power rule of logarithms, $\ln \left( \sqrt [ 3 ] { x ^ { 2 } - 3 } \right)$ becomes $\frac { 1 } { 3 } \ln \left( x ^ { 2 } - 3 \right)$, leading to the final expression $\ln x - \frac { 1 } { 3 } \ln \left( x ^ { 2 } - 3 \right)$.

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