I am reading through a first course in differential geometry, and would like some clarification on something. I am aware of the exponential map mapping from the lie algebra to the lie group, close to the identity (more precisely, to the connected component containing the identity). And the logarithm chart is its inverse on a suitably small open neighborhood of the identity of the lie group. I believe this is only defined for matrix lie groups.
In these notes, the first mention of lie groups also talks about logarithm and exponential maps. But not as maps between lie groups and lie algebras. Rather, the logarithm is used as the coordinate chart from the matrix lie group close to the identity, to a matrix lie algebra which can easily be identified with some subset of $\mathbb{R}^n$.
I feel like the notions are getting mixed up in these notes though. Take for instance theorem 1.7, and specifically the continuation of the prof at the very top of page 7. In order to show the closure of the lie algebra under, we show that the commutator of two elements of the lie aglebrga is in the image of the log chart, (albeit it is multiplied by t^2 rather than t so strictly it is not present to linear order)