  This paper continues the authors' program of studying mirror symmetry via log
geometry and toric degenerations, relating affine manifolds with singularities,
log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus
of this paper is the calculation of the cohomology of a Calabi-Yau variety
associated to a given affine manifold with singularities B. We show that the
Dolbeault cohomology groups of the Calabi-Yau associated to B are described in
terms of some cohomology groups of sheaves on B, as expected. This is proved
first by calculating the log de Rham and log Dolbeault cohomology groups on the
log Calabi-Yau space associated to B, and then proving a base-change theorem
for cohomology in our logarithmic setting.
  As applications, this shows that our mirror symmetry construction via
Legendre duality of affine manifolds results in the usual interchange of Hodge
numbers expected in mirror symmetry, and gives an explicit description of the
monodromy of a smoothing.
