$p$-adic logarithm (awaiting review)

Let $K$ be a finite extension of $\Q_p$ for some prime $p$, and $$ S=\{z \in K : |z|_p < 1\}$$ where $|\cdot|_p$ is an abaolute value on $K$ normalized so that $|p|_p=1/p$. Then the $p$-adic logarithm $ \log_p:S\to K$ is given by $$ \log_p(1+z) = \sum_{n=1}^{\infty}(-1)^n \frac{z^n}{n}.$$ One can extend $\log_p$ to $K$ by defining $\log_p(p)=0$ and $\log_p(\zeta)=0$ where $\zeta$ is any root of unity.

If $\mathcal{O}_K$ is the ring of integers of $K$ and $M$ is the unique maximal ideal of $\mathcal{O}_K$, then the image of the $p$-adic logarithm, $ \log_p(1+M) = M$ provided $e<p-1$ where $e$ is the ramification index for $K/\Q_p$.