Answer
The value of x is 16.
Work Step by Step
${{\log }_{4}}x-{{\log }_{4}}\left( x-15 \right)=2$
Simplify the logarithm ${{\log }_{4}}x-{{\log }_{4}}\left( x-15 \right)=2$ as follows.
${{\log }_{4}}x-{{\log }_{4}}\left( x-15 \right)=2$
Apply the quotient rule for logarithms,
${{\log }_{4}}\frac{x}{x-15}=2$
Use the fact that the expression ${{\log }_{a}}x=m\text{ is equivalent to }{{a}^{m}}=x$. Therefore,
$\begin{align}
& \frac{x}{x-15}={{4}^{2}} \\
& x=16\left( x-15 \right)
\end{align}$
Further simplify as follows.
$\begin{align}
& x=16x-240 \\
& 15x=240 \\
& x=\frac{240}{15} \\
& x=16
\end{align}$
Check:
Substitute $x=16$ in the given equation.
$\begin{align}
{{\log }_{4}}\left( 16 \right)-{{\log }_{4}}\left( 16-15 \right)\overset{?}{\mathop{=}}\,2 & \\
{{\log }_{4}}\left( 16 \right)-{{\log }_{4}}\left( 1 \right)\overset{?}{\mathop{=}}\,2 & \\
{{\log }_{4}}{{4}^{2}}-{{\log }_{4}}\left( 1 \right)\overset{?}{\mathop{=}}\,2 & \\
\end{align}$
Use power rule and solve.
$\begin{align}
{{\log }_{4}}{{4}^{2}}-{{\log }_{4}}\left( 1 \right)\overset{?}{\mathop{=}}\,2 & \\
2{{\log }_{4}}4-{{\log }_{4}}\left( 1 \right)\overset{?}{\mathop{=}}\,2 & \\
2-0\overset{?}{\mathop{=}}\,2 & \\
2=2 & \\
\end{align}$
Thus, the obtained solution is correct.
