We explore the relation between deformations of a generically log smooth family and deformations of the generically log smooth family together with a line bundle. The main result which we prove in this chapter is that in the log toroidal log Calabi–Yau case, these deformations are unobstructed. This gives the version of the logarithmic Bogomolov–Tian–Todorov theorem for pairs consisting of a log Calabi–Yau log toroidal family and a line bundle.