This chapter is the first part of our discussion of homological aspects of logarithmic deformation theory. A classical deformation problem is controlled by a dg Lie algebra, and deforming means solving the Maurer–Cartan equation. A logarithmic deformation problem is controlled by a curved Lie algebra instead. The presence of curvature forces us to modify the Maurer–Cartan equation into the “classical extended Maurer–Cartan equation,” whose solutions then correspond to logarithmic deformations. To solve the extended Maurer–Cartan equation, we consider it within the larger curved Gerstenhaber calculus, which provides additional operations to manipulate potential solutions. In the Calabi–Yau case, this can be further extended to a curved Batalin–Vilkovisky calculus. In any curved Batalin–Vilkovisky calculus satisfying suitable homological conditions, solutions of the extended Maurer–Cartan equation are unobstructed. This is the abstract unobstructedness theorem. We also discuss a version that incorporates deformations of the volume form, the “semi-classical extended Maurer–Cartan equation.”