This paper lays the foundations of a program to study mirror
symmetry by studying the log structures of Illusie-Fontaine
and Kato on degenerations of Calabi-Yau manifolds. The basic
idea is that one can associate to certain sorts of degenerations
of Calabi-Yau manifolds a log Calabi-Yau space, which is a log
structure on the degenerate fibre. The log CY space captures
essentially all the information of the degeneration, and hence all
mirror statements for the "large complex structure limit" given by
the degeneration can already be derived from the log CY space.
In this paper we begin by discussing affine manifolds with singularities.
Given such an affine manifold along with a polyhedral
decomposition, we show how to construct a scheme consisting of
a union of toric varieties. In certain non-degenerate cases, we can
also construct log structures on these schemes. Conversely, given
certain sorts of degenerations, one can build an affine manifold
with singularities structure on the dual intersection complex of
the degeneration. Mirror symmetry is then obtained as a discrete
Legendre transform on these affine manifolds, thus providing an
algebro-geometrization of the Strominger-Yau-Zaslow conjecture.
The deepest result of this paper shows an isomorphism between
log complex moduli of a log CY space and log Kähler moduli of
its mirror.