Types and nature of singularities of f(z)=1/z ln(1-z)

Question

Consider the complex function f(z) = (1/z)×ln(1-z) , It seems like having a removable singularity (because,while comparing with corresponding real function,the limit exists.)
The function has a branch point at z=1. Apart from my above findings, what more singularities are there?.

Answer 1

If $\ln z$ is the principal branch of logarithm then you have to remove the entire $[1,\infty)$ to get an analytic function. The function is not even continuous at points of this interval. Once you remove this part of the real axis you get an analytic function; the singularity at $0$ is actually removable.