The logarithm near the identity and local inverse of exp

1. $\sum_k\frac{\|X\|^k}k=-\ln(1-\|X\|)<\infty$ for $\|X\|<1$.

2. The identity $e^{\log(I+X)}=I+X$. Let $x$ be a formal variable. In $\mathbb{Q}[[x]]$ one has $\exp(\log(1+x))=1+x$ (a known identity of formal power series). Substituting $X$ for $x$ is legitimate when the double series of norms converges: the norm-majorant is
$$\sum_{m\ge0}\frac1{m!}\Bigl(\sum_{k\ge1}\frac{\|X\|^k}k\Bigr)^m=\exp\bigl(-\ln(1-\|X\|)\bigr)=\frac1{1-\|X\|}<\infty.$$
Absolute convergence allows rearranging the terms into powers of $X$, and the coefficients are those of the formal identity. Hence $e^{\log(I+X)}=I+X$.

Similarly, for $\|A\|<\ln2$ put $X=e^A-I$. Then $\|X\|\le e^{\|A\|}-1<1$ and the formal identity $\log(\exp x)=x$ (that is, $\log(1+(e^x-1))=x$) gives $\log(e^A)=A$ once the double series converges absolutely: its majorant is
$$\sum_k\frac{(e^{\|A\|}-1)^k}k=-\ln\bigl(2-e^{\|A\|}\bigr)<\infty\quad\text{exactly when }\|A\|<\ln2.$$

3. Local homeomorphism. Let $U=\{I+X:\|X\|<1,\ \|\log(I+X)\|<\ln2\}$. It is open, as the preimage of $B(0,\ln2)$ under the continuous map $\log$ on the open ball $B(I,1)$ (intersected with that ball).
- If $\|A\|<\ln2$, then $\|e^A-I\|<1$ and $\log e^A=A$ has norm $<\ln2$, so $e^A\in U$.
- If $I+X\in U$ then $A=\log(I+X)\in B(0,\ln2)$ and $e^A=I+X$.

So $\exp:B(0,\ln2)\to U$ is a bijection with continuous inverse $\log$, hence a homeomorphism. Finally $U\supset B(I,\frac12)$: if $\|X\|<\frac12$ then $\|\log(I+X)\|\le-\ln(1-\|X\|)<\ln2$.

Remark. This shows that $\exp$ is a local homeomorphism at $0$: every matrix close to $I$ has a (unique small) logarithm. It is the first step in proving that closed subgroups of $GL_n$ are Lie groups.