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) ${\left(\frac{(\ln x{)}^2}{\ln x+\ln 3}\right)}^{1/5}$

Properties

Characteristics or rules that define how operations are performed on numbers and variables, including distributive, associative, and commutative properties.

Logarithms

Mathematical functions that determine the exponent or power to which a base number is raised to obtain a certain value, often written as \(\log_b(x)\).

Expand

To increase in size, number, or importance, or to make something increase in this way.