Let $G \leq GL(n,\mathbb C)$ be a matrix Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$. Let $U \subset \mathfrak g$ be an open neighbourhood of $0$, let $V \subset G$ be an open neighbourhood of $I$, and suppose that
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\begin{align*}
\exp|_U:U\to V
\end{align*}
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is a diffeomorphism with inverse
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\begin{align*}
\log:V\to U.
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For each $g\in G$, define the translated logarithm chart