  In a previous paper [FT1], for any logarithmic symplectic pair (X,D) of a
symplectic manifold X and a simple normal crossings symplectic divisor D, we
introduced the notion of log pseudo-holomorphic curve and proved a compactness
theorem for the moduli spaces of stable log curves. In this paper, we introduce
a natural set up for studying the deformation theory of log (and relative)
curves. As a result, we obtain a logarithmic analog of the space of Ruan-Tian
perturbations for these moduli spaces. For a generic compatible pair of an
almost complex structure and a log perturbation term, we prove that the
subspace of simple maps in each stratum is cut transversely. Such perturbations
enable a geometric construction of Gromov-Witten type invariants for certain
semi-positive pairs (X, D) in arbitrary genera. In future works, we will use
local perturbations and a gluing theorem to construct log Gromov- Witten
invariants of arbitrary such pair (X, D).
