Question 3: Expand the following using laws of logarithms:
\((ii)\) \(\quad\) \(log_{5}\sqrt{8a^{6}}\)
Solution:
\[
\begin{align*}
&=\log_{5}\sqrt{8a^{6}}\\
&=\log_{5}\left( 8a^{6}\right)^{\frac{1}{2}}\\
&=\frac{1}{2}\log_{5}\left( 8a^{6}\right)\\
&=\frac{1}{2}\left(\log_{5} 8+\log_{5} a^{6}\right)\\
&=\frac{1}{2}\left(\log_{5} 2^{3} +6\log_{5} a\right)\\
&=\frac{1}{2}( 3\log_{5} 2 +6\log_{5} a)\\
&=\frac{3}{2}\log_{5} 2 +\frac{6}{2}\log_{5} a\\
&=\frac{3}{2}\log_{5} 2 +3\log_{5} a\\
\end{align*}
\]
——– Alternate Method ——–
\[
\begin{align*}
&=\log_{5}\sqrt{8a^{6}}\\
&=\log_{5}\left( 8a^{6}\right)^{\frac{1}{2}}\\
&=\log_{5}\left( 2^{3} a^{6}\right)^{\frac{1}{2}}\\
&=\log_{5}\left( 2^{\large{\frac{3}{2}}} a^{\large{\frac{6}{2}}}\right)\\
&=\log_{5}\left( 2^{\large{\frac{3}{2}}} a^{3}\right)\\
&=\log_{5} 2^{\large{\frac{3}{2}}} +\log_{5} a^{3}\\
&=\frac{3}{2}\log_{5} 2 +3\log_{5} a\\
\end{align*}
\]