Answer
$$\log (\frac{x^2-4}{\sqrt {x^2+4}})$$
Work Step by Step
$Combine$ $into$ $a$ $single$ $logarithm:$
$\log (x-2)$ + $\log (x+2)$ - $\frac{1}{2}\log (x^2+4)$
Use the Third Law of Logarithms for $\frac{1}{2}\log (x^2+4)$
$\frac{1}{2}\log (x^2+4)$ = $\log (x^2+4)^{\frac{1}{2}}$
$\log (x-2)$ + $\log (x+2)$ - $\log (x^2+4)^{\frac{1}{2}}$
Use the First Law of Logarithms for $\log (x-2)$ + $\log (x+2)$
$\log (x-2)$ + $\log (x+2)$ = $\log ((x-2)(x+2))$
$\log ((x-2)(x+2))$ - $\log (x^2+4)^{\frac{1}{2}}$
Use the Second Law of Logarithms
$\log ((x-2)(x+2))$ - $\log (x^2+4)^{\frac{1}{2}}$ = $\log (\frac{(x-2)(x+2)}{(x^2+4)^{\frac{1}{2}}})$
$$\log (\frac{(x-2)(x+2)}{(x^2+4)^{\frac{1}{2}}})$$
Simplify
$$\log (\frac{x^2-4}{\sqrt {x^2+4}})$$