Answer
The solution of the equation ${{\log }_{4}}\left( {{x}^{2}}-9 \right)-{{\log }_{4}}\left( x+3 \right)=3$ is $67$.
Work Step by Step
Consider the provided equation,
${{\log }_{4}}\left( {{x}^{2}}-9 \right)-{{\log }_{4}}\left( x+3 \right)=3$
Using, for $ a>0$ and $ b>0$, ${{\log }_{b}}a-{{\log }_{b}}c={{\log }_{b}}\left( \frac{a}{c} \right)$
$\begin{align}
& {{\log }_{4}}\left( \frac{{{x}^{2}}-9}{x+3} \right)=3 \\
& {{\log }_{4}}\left( \frac{\left( x-3 \right)\left( x+3 \right)}{x+3} \right)=3 \\
& {{\log }_{4}}\left( x-3 \right)=3
\end{align}$
Using,if for $ a>0$ and $ b>0$, ${{\log }_{b}}a=c $, then ${{b}^{c}}=a $
$\begin{align}
& {{\log }_{4}}\left( x-3 \right)=3 \\
& x-3={{4}^{3}} \\
& x-3=64 \\
& x=67
\end{align}$
Thus, the solution of the equation ${{\log }_{4}}\left( {{x}^{2}}-9 \right)-{{\log }_{4}}\left( x+3 \right)=3$ is $ x=67$
