  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
$\Phi$ to derive a formula for $\log(\Phi)$ in terms of MZV's (multiple zeta
values).
