Radius of convergence of power series of complex $\log$ Let $f(z) = \log(z)$ for $z\in \Bbb{C}\setminus (-\infty,0]$.
Since $f$ is holomorphic on its domain, we know it has a power series development about each point $z_0\in \Bbb{C}\setminus (-\infty,0]$. We also that its radius of convergence will be at least the distance of $z_0$ to the nearest singularity.
How can we determine the exact radius of convergence?

The radius of convergence will be $|z_0|$. 
It is clear that it can't be greater that $|z_0|$, since $\log z$ can't be defined  (holomorphically) on any neighbourhood of the origin.
On the other hand, the disc $D = \{ z : |z-z_0| < |z_0| \}$ is simply connected and doesn't contain $0$. Hence we can find a branch of the complex logarithm (that agrees with your choice of $\log$ near $z_0$. Taylor's theorem shows that the radius of convergence must be at least (and thus exactly) $|z_0|$.
Note, however, that the power series you get does not necessarily coincide with the principle branch of $\log z$ on all of $D$.
