We lay the foundations for transforming a logarithmic deformation problem into a deformation problem of algebraic structures, most importantly into a deformation problem of Gerstenhaber calculi. To this end, we fix a framework to study families of algebraic structures on schemes and their deformations. We do this for all algebraic structures of interest, such as Lie–Rinehart algebras and Gerstenhaber calculi, at once. We also describe the transition from a logarithmic deformation problem to a deformation problem of Gerstenhaber calculi and show the equivalence of the resulting deformation functors.