Answer
The solution set is $\{2\}$.
Work Step by Step
The given equation is
$\log _4(x+2) -\log _4(x-1)=1$
Use quotient rule.
$\log _4 \left [\frac{x+2}{x-1} \right]=1$
Rewrite in exponential form.
$\frac{x+2}{x-1}= 4^1$
Multiply both sides by $x-1$
$(x-1)\cdot \frac{x+2}{x-1}= (x-1)\cdot 4$
Use the distributive property on the right side.
$x+2=4x-4$
Add $4-x$ to both sides.
$x+2+4-x=4x-4+4-x$
Simplify.
$6=3x$
Divide both sides by $3$.
$\frac{6}{3}=\frac{3x}{3}$
Simplify.
$2=x$
The solution is $x=2$.
