Answer
$\log_5\frac{\sqrt{xy}}{(x+1)^2}$
Work Step by Step
$$A=\frac{1}{2}(\log_5x+\log_5y)-2\log_5(x+1)$$
First, we know from the Product Rule that $$\log_bM+\log_bN=\log_bMN$$ ($M, N, b\in R, M\gt0, N\gt0, b\gt0, b\ne1$)
Apply it to $\log_5x+\log_5y$, we have
$$A=\frac{1}{2}\log_5(xy)-2\log_5(x+1)$$
Now, from the Power Rule that $$p\log_bM=\log_bM^p$$ ($M, b, p\in R, M\gt0, b\gt0, b\ne1$), we also can deduce
$$A=\log_5(xy)^{1/2}-\log_5(x+1)^2$$ $$A=\log_5\sqrt{xy}-\log_5(x+1)^2$$
Finally, we know the Quotient Rule, which states $$\log_b M-\log_bN=\log_b\frac{M}{N}$$ ($M, N, b\in R, M\gt0, N\gt0, b\gt0, b\ne1$)
That means,
$$A=\log_5\frac{\sqrt{xy}}{(x+1)^2}$$
