Question 2: Write the following as a single logarithm:
\((iii)\) \(\quad\) \(\log_5 b^2.\log_a 5^3 \)
Solution:
\[
\begin{align*}
&=\log_{5} b^{2} \cdot \log_{a} 5^{3}\\
&=2\log_{5} b \cdot 3\log_{a} 5\\
&=( 2\times 3) \cdot \frac{\log b}{\log 5} \cdot \frac{\log 5}{\log a}\\
&=6\cdot \frac{\log b}{\log a}\\
&=6\cdot \log_{a} b
\end{align*}
\]