The expression is
$$\log\left(\left| \frac{z+ia}{z-ia}\right|\right) $$
Here, $a>0$ is real, $i$ is the imaginary unit, and $z=x+iy$. The expression comes from extracting the imaginary part of an expression so it should end up being purely real, however I can't manage to remove the imaginary unit from any of my expressions. I know that it should end up being something like:
$$\frac{1}{2} \log\left( \frac{x^2 + (y+a)^2}{x^2 + (y-a)^2}\right)$$
Note that $|z|=|x+iy|=\sqrt{x^2+y^2}$ when both $x$ and $y$ are assumed to be real numbers. Therefore, we have
$$\left|\frac{z+ia}{z-ia}\right|=\sqrt{\frac{x^2+(y+a)^2}{x^2+(y-a)^2}}$$
Can you finish now?