Answer
$x=\displaystyle \frac{17}{4}$
Work Step by Step
First, the solutions must satisfy $\left\{\begin{array}{l}
x+7\gt 0\\
x-2\gt 0
\end{array}\right.\quad \Rightarrow x\gt 2\qquad (*)$
in order for the equation to be defined.
LHS: Apply$ \displaystyle \quad\log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$
$\displaystyle \log_{6}[\frac{x+7}{x-2}]=\log_{6}5$
... apply the principle of logarithmic equality
$\displaystyle \frac{x+7}{x-2}=5$
$x+7=5(x-2)$
$x+7=5x-10$
$-4x=-17$
$x=\displaystyle \frac{17}{4}\qquad $... satisfies (*), and is a valid solution.
