Branch of logarithm which is real when z>0 I am familiar with the complex logarithm and its branches, but still this confuses me. I read this in a textbook:
"For complex $z\neq 0, log(z)$ denotes that branch of the logarithm which is real when $z > 0$. 
What does this mean?

Let $\hbox{Log} z=\ln|z|+i\hbox{Arg}z$ be the principal branch of the logarithm,
that corresponds to a cut along  the negative real numbers. Now, consider any branch of $log$ that is defined on a connected neighborhood of $\Bbb{R}^+=(0,+\infty)$. Clearly,
$x\mapsto \varphi(x)=log(x)-\hbox{Log}(x)$ is a continuous function on  $\Bbb{R}^+$ that satisfies $\exp(\varphi(x))=1$ so, $\varphi$ takes its values in $2\pi i\Bbb{Z}$, and since it is continuous, it must be constant. Thus, there is $k\in\Bbb{Z}$ such that
$log(z)=\hbox{Log }{z}+2i\pi k$ and this happens  on the largest connected open set that contains
 $\Bbb{R}^+=(0,+\infty)$, and on which both functions are analytic (by analytic continuation). 
Thus, the statement says that we choose the logarithmic function the corresponds to $k=0$ because otherwise $log(z)$ would not be real for real $z$.
