Lazard correspondence establishes a 1-isomorphism between Lazard Lie group and Lazard Lie ring

Statement
Suppose $$G$$ is a Lazard Lie group and $$L$$ is its Lazard Lie ring with $$\log:G \to L$$ the logarithm map and $$\exp:L \to G$$ the exponential map. (These are both bijections, and are inverses of each other).

Then, $$\log$$ and $$\exp$$ are 1-isomorphisms, i.e., they are isomorphisms when restricted to cyclic subgroups.

Facts used

 * 1) uses::Logarithm map from Lazard Lie group to its Lazard Lie ring is a quasihomomorphism

Proof outline
This follows directly from fact (1), and the fact that the logarithm map is bijective.