Single atlas implies global normal coordinates?

Let $M$ be a class $C^k$-Riemannian manifold and suppose there exists an atlas $\langle U,\psi\rangle$ for $M$ containing only one global chart.

• Does this imply that the Riemmanian $Log_p\,(:=Exp_p^{-1})$ maps are defined globally?

Ie: for every point $p\in M$ $Dom(Log_p)=M$?

• If so can we give a simple and explicit description of these coordinates in terms of the map $\psi$?