Let $n \in \mathbb N$, let $M(n,\mathbb C)$ denote the complex algebra of $n \times n$ matrices with identity matrix $I_n$, and let $GL(n,\mathbb C)$ denote the group of invertible elements of $M(n,\mathbb C)$. Let $\|\cdot\|_*$ be a submultiplicative matrix norm on $M(n,\mathbb C)$, meaning that $\|AB\|_* \leq \|A\|_*\|B\|_*$ for all $A,B \in M(n,\mathbb C)$. Define the matrix exponential map

\begin{align*} \exp: M(n,\mathbb C) \to M(n,\mathbb C),\qquad \exp X:=\sum_{k=0}^{\infty}\frac{X^k}{k!}. \end{align*}

Let

\begin{align*} \mathcal D_{\log}:=\{Y \in M(n,\mathbb C):\|Y-I_n\|_*<1\}. \end{align*}

Define the power-series logarithm map

\begin{align*} \log: \mathcal D_{\log} \to M(n,\mathbb C),\qquad \log Y:=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(Y-I_n)^k}{k}. \end{align*}

Then there exist open neighbourhoods $U \subset M(n,\mathbb C)$ of $0$ and $V \subset GL(n,\mathbb C)$ of $I_n$, with

\begin{align*} V \subset \mathcal D_{\log}, \end{align*}

such that

\begin{align*} \log(\exp X)=X \end{align*}

for every $X \in U$, and

\begin{align*} \exp(\log Y)=Y \end{align*}

for every $Y \in V$.