Answer
$\log_5{\frac{\sqrt{xy}}{(x+1)^2}}$
Work Step by Step
The product rule for logarithms says that $\log_b{MN}=\log_bM+\log_bN$ i.e. the logarithm of a product is the sum of the logarithms.
The quotient rule for logarithms says that $\log_b{\frac{M}{N}}=\log_bM-\log_bN$ i.e. the logarithm of a quotient is the difference of the logarithms.
The power rule for logarithms says that $\log_b{M^p}=p\log_bM$ i.e. the logarithm of a number with an exponent is the exponent times the logarithm of the number.
$\log_ba=\frac{\log_ca}{\log_cb}$
Hence here: $\frac{1}{2}(\log_5 x+\log_5y)-2\log_5{(x+1)}=\frac{1}{2}\log_5{(xy)}-\log_5{(x+1)^2}=\log_5{(xy)^{\frac{1}{2}}}-\log_5{(x+1)^2}=\log_5{\frac{(xy)^{\frac{1}{2}}}{(x+1)^2}}=\log_5{\frac{\sqrt{xy}}{(x+1)^2}}$
