Answer
$\displaystyle \log_{5}\frac{\sqrt{xy}}{(x+1)^{2}}$
Work Step by Step
$\displaystyle \frac{1}{2}(\log_{5}x+\log_{5}y)-2\log_{5}(x+1)$
apply product rule for the first term and power rule for the second term:
$=\displaystyle \frac{1}{2}\log_{5}(xy)-\log_{5}(x+1)^{2}$
apply power rule for the first term:
$=\log_{5}(xy)^{1/2}-\log_{5}(x+1)^{2}$
apply the quotient rule:
$=\displaystyle \log_{5}\frac{(xy)^{1/2}}{(x+1)^{2}}\quad $...apply: $ a^{1/2}=\sqrt{a}$
$=\displaystyle \log_{5}\frac{\sqrt{xy}}{(x+1)^{2}}$
