## College Algebra (6th Edition)

$\log_5\frac{\sqrt{xy}}{(x+1)^2}$
$$A=\frac{1}{2}(\log_5x+\log_5y)-2\log_5(x+1)$$ 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, $$A=\log_5\frac{\sqrt{xy}}{(x+1)^2}$$