Let $M$ be an element of a Banach algebra $\mathbb{L}$ such that its spectrum $\sigma(M)$ does not contain $0$ and $0$ is connected to $\infty$ by a curve that lies in the resolvent $\rho(M)$.
Then, how do I show that the logarithm of $M$, $log (M)$ can be defined such that $e^{ log(M)} = M$? I think I have to define the logarithm in terms of an infinite sequence. But I cannot see the need of the condition " $0$ is connected to $\infty$ by a curve that lies in the resolvent $\rho(M)$".
Could anyone help me how to rigorously define $log(M)$?