Does the matrix logarithm always converge for exponential matrices? We have defined functions on square matrices $X$:
$$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$
and 
$$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$
The exponential converges for all matrices, but the logarithm converges for all $X$ with norm (maximum eigenvalue) less than or equal to 1. The two can be shown to be inverses of each other whenever the logarithm is defined. We thus have $$e^{log(X)} = X$$ whenever $log(X)$ is defined. But also $$log(e^X)=X$$ whenever $log(e^X)$ is defined. 
My question is, is log defined for all matrices which can be written in the form $e^X$?

No, just consider the $1 \times 1$ case, i.e. real numbers. Let $X=10$ and the series doesn't converge. 
