Answer
$x=\dfrac{43}{21}$
Work Step by Step
Using properties of logarithms and since $\log_b x=y$ implies $b^y=x$, then,
\begin{array}{l}
\log_4 (x+1)-\log_4 (x-2)=3
\\\\
\log_4 \dfrac{x+1}{x-2}=3
\\\\
\dfrac{x+1}{x-2}=4^3
\\\\
x+1=64(x-2)
\\\\
x+1=64x-128
\\\\
x-64x=-128-1
\\\\
-63x=-129
\\\\
x=\dfrac{-129}{-63}
\\\\
x=\dfrac{43}{21}
.\end{array}
