Logarithm map for Lie groups other than Matrix Lie groups

Question

Hello Lie group experts!

My question is simple: Is there a construct for logarithm map of non-Matrix Lie groups? Besides; I believe it exists, because $exp(t.X) \in G \neq GL(n, F)$ being diffeomorphism would not be possible otherwise, where $X \in \mathfrak{g}\neq gl(n, F)$ and $t \in \mathbb{R}$. Furthermore, may I assume that (almost) all theorems applied to Matrix-Lie groups using the relationship between log and $exp$ maps are still valid? For instance:

For any $B=exp(X) \in V_\varepsilon $, $\exists!$ square root of B called $C=exp(\frac{1}{2}X)$; s.t. $C \in V_{\varepsilon}$ and $V_\varepsilon \subset GL(n, F)$ containing $e$.

This theorem is based on the relationship between log and $exp$ maps for Matrix-Lie groups. However, I am not sure whether it is still valid for any Lie group $G$. At least, we may assume (if necessary) that $G$ is either:

1. Compact, or
2. Non-compact, but nilpotent.