Answer
$\log{\left(\dfrac{x^2-4}{\sqrt{x^2+4}}\right)}$
Work Step by Step
RECALL:
(1) $n \cdot \log_a{P}=\log_a{(P^n)}$
(2) $\log_a{P} + \log_a{Q}=\log_a{(PQ)}$
(3) $\log_a{P} - \log_a{Q}=\log_a{(\frac{P}{Q})}$
(4) $a^m \cdot a^n=a^{m+n}$
Use rule (1) above to obtain
$\log{(x-2)}+\log{(x+2)}-\frac{1}{2}\log{(x^2+4)}
\\=\log{(x-2)}+\log{(x+2)}-\log{(x^2+4)^{\frac{1}{2}}}$
Use rule (2) above to obtain:
$=\log{[(x-2)(x+2)]}-\log{x^2+4)^{\frac{1}{2}}}$
Since $(a-b)(a+b)=a^2-b^2$, then the expression above simplifies to:
$=\log{(x^2-4)} - \log{(x^2+4)^{\frac{1}{2}}}$
Use rule (3) above to obtain:
$=\log{\left(\dfrac{x^2-4}{(x^2+4)^{\frac{1}{2}}}\right)}$
Use the rule $a^{\frac{1}{2}}=\sqrt{a}$ to obtain:
$=\log{\left(\dfrac{x^2-4}{\sqrt{x^2+4}}\right)}$
