Question 8: Expand the following using laws of logarithms:
\((ii)\) \(\quad\) \(log_{3} \ \sqrt[6]{m^{5} n^{3}} \)
Solution:
\[
\begin{align*}
&=\log_{3}\sqrt[3]{m^{5} n^{3}}\\
&=\log_{3}\left( m^{5} n^{3}\right)^{\frac{1}{6}}\\
&=\frac{1}{6}\log_{3}\left( m^{5} n^{3}\right)\\
&=\frac{1}{6}\left[\log_{3} m^{5} +\log_{3} n^{3}\right]\\
&=\frac{1}{6}[ 5\log_{3} m +3\log_{3} n]\\
\end{align*}
\]