Answer
$x=8$
Work Step by Step
First, the solutions must satisfy $\left\{\begin{array}{l}
x+2\gt 0\\
x-7\gt 0
\end{array}\right.\quad \Rightarrow x\gt 7\qquad (*)$
in order for the equation to be defined.
On the RHS, apply$ \quad\log_{a}(MN)=\log_{a}M+\log_{a}N$
$\log_{4}[(x+2)(x-7)]=\log_{4}10$
... apply the principle of logarithmic equality
$(x+2)(x-7)=10$
$x^{2}-5x-14=10$
$ x^{2}-5x-24=0\quad$to factor, find factors of $-24$ with sum $-5$
... these are $-8$ and $+3$
$(x+3)(x-8)=0$
Possible solutions:
$ x=-3\qquad$... does not satisfy (*), not a solution.
$x=8\qquad $... satisfies (*), and is a valid solution.
