A Formula for the Logarithm of the KZ Associator

We prove that the logarithm of a group-like element in a free algebra coincides with its image by a certain linear map. We use this result and the formula of Le and Murakami for the Knizhnik-Zamolodchikov (KZ) associator $\mathrm{\Phi }$ to derive a formula for $\log (\mathrm{\Phi })$ in terms of MZV’s (multiple zeta values).