The exponential map goes from the tangent space to the manifold, and the log map goes back.
In reading, however, I get the impression that people use the "exponential map" as a term for the overall correspondence, or for travel in either direction. For example, people rarely say "using the log map".
Is this impression correct?
On
Let $ (M,g) $ be a Riemannian manifold that is geodesically complete. By the Hopf-Rinow Theorem, the exponential mapping at each $ p \in M $ is then defined on all of $ {T_{p}}(M) $. Let $ \exp_{p} $ be the exponential mapping at $ p $ and $ U \subseteq {T_{p}}(M) $ be a sufficiently small open neighborhood of $ 0_{{T_{p}}(M)} $ such that $ \exp_{p}|_{U}: U \rightarrow (\exp_{p}|_{U})[U] $ is a diffeomorphism. Then it is common practice by differential geometers to denote the inverse of $ \exp_{p}|_{U} $ simply by $ (\exp_{p}|_{U})^{-1} $.
Some people use the notation ‘$ \log $’, but then, the domain and range must be specified.
Yes. I would never talk about the "log map." I do however at times refer to "the inverse of the exponential map'' or "the inverse exponential map."
Moreover, I have never heard a professional differential geometer talk about the "log map."