Answer
The solution set is $\{6\}$.
Work Step by Step
The given equation is
$\log _2(x+2) -\log _2(x-5)=3$
Use quotient rule.
$\log _2 \left [\frac{x+2}{x-5} \right]=3$
Rewrite in exponential form.
$\frac{x+2}{x-5}= 2^3$
Multiply both sides by $x-5$
$(x-5)\cdot \frac{x+2}{x-5}= (x-5)\cdot 8$
Use the distributive property on the right side.
$x+2=8x-40$
Add $40-x$ to both sides.
$x+2+40-x=8x-40+40-x$
Simplify.
$42=7x$
Divide both sides by $7$.
$\frac{42}{7}=\frac{7x}{7}$
Simplify.
$6=x$
The solution is $x=6$.
