  We introduce the notion of a logarithmic Landau-Ginzburg (log LG) model,
which is essentially given by equipping the central degenerate fiber of the
family of Landau-Ginzburg (LG) models mirror to a projective toric manifold
with a natural log structure. We show that the state space of the mirror log LG
model is naturally isomorphic to that of the original toric manifold. Following
Li-Li-Saito, we give a perturbative construction of primitive forms by studying
the deformation theory of such a log LG model, which involves both smoothing of
the central degenerate fiber and unfolding of the superpotential. This yields a
logarithmic Frobenius manifold structure on the base space of the universal
unfolding. The primitive forms and flat coordinates we obtained are computable
and closely related to the bulk-deformed Lagrangian Floer superpotential of a
projective toric manifold, at least in the semi-Fano case.
