I highly recommend this paper for a more in-depth treatment of exp/log map and the related representation of transformations:
A Micro Lie Theory by Joan Sola, Jeremie Deray, Dinesh Atchuthan https://arxiv.org/pdf/1812.01537.pdf
A Micro Lie Theory by Joan Sola, Jeremie Deray, Dinesh Atchuthan https://arxiv.org/pdf/1812.01537.pdf
This treatment is closer to the mathematician's than to the physicist's, but it still has some of the physics flavor about it—for example, it at least implicitly discusses Lie groups as if they come with a preferred action, which they need not—and, if you're not in this domain already, it's good to know what difficulties you can face later if you try to move between domains.
This is not always true, I think sl2(R) is an example of a connected, non compact group such that the exponential map is not surjective.
The Wikipedia page[1] mentions the issue briefly but I was curious of the counter-examples. IIUC, matrices in SL(2) with trace < -2 have two distinct eigenvalues, one of which is negative[2], and such matrices cannot be reached by exponentiating elements of the Lie algebra sl(2) (traceless matrices).
[1] https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#S...
[2] https://en.wikipedia.org/wiki/SL2(R)#Classification_of_eleme...