We describe how log structures can be modified by a section of a line bundle. This allows us to transform a log Fano deformation problem into a log Calabi–Yau deformation problem. As a consequence, we obtain a logarithmic Bogomolov–Tian–Todorov theorem for many log Fano and related situations. We apply the result to smooth normal crossing spaces with globally generated anti-dualizing sheaf and globally generated first tangent sheaf (in the sense of Lichtenbaum–Schlessinger).