Solution
= Solution
All values of the <complex logarithm> are
$$
w=\log|z|+i(\arg z+2\pi n),
\qquad n\in\mathbb Z.
$$
Because $x>0$, one may take the <complex argument> $\arg z=\arctan(y/x)$. Consequently
$$
\boxed{\operatorname{Re}w=\frac12\log(x^2+y^2),
\qquad
\operatorname{Im}w=\arctan\frac yx+2\pi n,quad n\in\mathbb Z}.
$$