Answer
$\frac{(\ln5)^2\ln3}{2}$
Work Step by Step
We begin with the iterated integral:
$$\int_{1}^{3}\int_1^5\frac{\ln y}{xy}\,dy\,dx$$
Let $u=\ln y$ and $du=\frac{1}{y}dy$
We can rewrite the integral as:
$$\int_{1}^{3}\int_0^{\ln5}\frac{u}{x}\,du\,dx$$
Solving, we get:
$$\int_{1}^{3}\bigg[\frac{u^2}{2x}\bigg]_{u=0}^{u=\ln5}dx\\
=\frac{(\ln5)^2}{2}\int_{1}^{3}\frac{1}{x}dx\\
=\frac{(\ln5)^2}{2}\bigg[ln{|x|}\bigg]_1^3\\
=\frac{(\ln5)^2\ln3}{2}$$
