Answer
The value of the given expression has no solution.
Work Step by Step
Consider the given equation,
${{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2$
Apply the quotient rule of logarithms:
$\begin{align}
& {{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2 \\
& {{\log }_{3}}\frac{\left( x-1 \right)}{\left( x+2 \right)}=2
\end{align}$
The provided equation can be written as
$\begin{align}
& \frac{\left( x-1 \right)}{\left( x+2 \right)}={{3}^{2}} \\
& \frac{\left( x-1 \right)}{\left( x+2 \right)}=9 \\
& \left( x-1 \right)=9\left( x+2 \right) \\
\end{align}$
$ x-1=9x+18$
Now, add to both sides $-x-18$:
$\begin{align}
& x-1-x-18=9x+18-x-18 \\
& 8x=-19 \\
& x=\frac{-19}{8} \\
\end{align}$
Here $ x=\frac{-19}{8}$ is not the solution of the given expression because
$\begin{align}
& {{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2 \\
& {{\log }_{3}}\left( \frac{-19}{8}-1 \right)-{{\log }_{3}}\left( \frac{-19}{8}+2 \right)=2 \\
& {{\log }_{3}}\left( \frac{-19}{8}-1 \right)-{{\log }_{3}}\left( \frac{-19}{8}+2 \right)=2 \\
& {{\log }_{3}}\left( -3.375 \right)-{{\log }_{3}}\left( -0.375 \right)=2
\end{align}$
By definition of ${{\log }_{a}}b $, $ b>0$ but $-3.375$ and $-0.375$ is not greater than zero.
Thus, $ x=\frac{-19}{8}$ does not satisfy the equation ${{\log }_{3}}\left( x-1 \right)-{{\log }_{3}}\left( x+2 \right)=2$
Therefore, there is no solution of the given expression.
