[tex]\bf log_{{ a}}(xy)\implies log_{{ a}}(x)+log_{{ a}}(y)
\\ \quad \\\\
% Logarithm of exponentials
log_{{ a}}\left( x^{{ b}} \right)\implies {{ b}}\cdot log_{{ a}}(x)\\\\
and\qquad a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad
\sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}\\\\
-----------------------------\\\\[/tex]
[tex]\bf ln\left( \sqrt[4]{xy} \right)\implies ln\left[ (xy)^{\frac{1}{4}} \right]\implies ln\left[ x^{\frac{1}{4}} y^{\frac{1}{4}} \right]
\\\\\\
ln\left( x^{\frac{1}{4}} \right)+ln\left( y^{\frac{1}{4}} \right)\implies \frac{1}{4}ln(x)+\frac{1}{4}ln(y)[/tex]
