Use the properties of logarithms to expand $\ln \sqrt [ 5 ] { \frac { x ^ { 2 } } { x + 3 } }$ .
A) $( \ln x ) ^ { 2 / 5 } - \ln x \cdot \ln 3$
B) $- 5 ( 2 \ln x - \ln ( x + 3 ) )$
C) $\frac { 1 } { 5 } ( 2 \ln x - \ln x \cdot \ln 3 )$
D) $\frac { 2 } { 5 } \ln x - \frac { 1 } { 5 } \ln ( x + 3 )$
E) $\left( \frac { ( \ln x ) ^ { 2 } } { \ln x + \ln 3 } \right) ^ { 1 / 5 }$

Question

Quizplus · Accepted Answer

The expression can be expanded using the properties of logarithms: $\ln \sqrt[5]{\frac{x^2}{x+3}}$ simplifies to $\frac{1}{5}\ln\left(\frac{x^2}{x+3}\right)$, which further simplifies to $\frac{1}{5}(2\ln x - \ln(x+3))$, matching choice D.

Use the Properties of Logarithms to Expand ln⁡x2x+35\ln \sqrt [ 5 ] { \frac { x ^ { 2 } } { x + 3 } }ln5x+3x2​​

Use the Properties of Logarithms to Expand $\ln \sqrt [ 5 ] { \frac { x ^ { 2 } } { x + 3 } }$