Logarithm map from Lazard Lie group to its Lazard Lie ring is a quasihomomorphism

Statement
Let $$G$$ be a Lazard Lie group and $$L$$ be its Lazard Lie ring. Suppose $$\log:G \to L$$ is the logarithm map, i.e., one direction of the bijection between $$G$$ and $$L$$. This map is a quasihomomorphism of groups.