Abstract We establish a patchworking theorem à la Viro for the Log-critical locus of algebraic curves in ( ℂ ∗ ) 2 {(\mathbb{C}^{\ast})^{2}} . As an application, we prove the existence of projective curves of arbitrary degree with smooth connected Log-critical locus. To prove our patchworking theorem, we study the behaviour of Log-inflection points along families of curves defined by Viro polynomials. In particular, we prove a generalisation of a theorem of Mikhalkin and the second author on the tropical limit of Log-inflection points.