  A great number of theoretical results are known about log Gromov-Witten
invariants, but few calculations are worked out. In this paper we restrict to
surfaces and to genus 0 stable log maps of maximal tangency. We ask how various
natural components of the moduli space contribute to the log Gromov-Witten
invariants. The first such calculation by Gross-Pandharipande-Siebert deals
with multiple covers over rigid curves in the log Calabi-Yau setting. As a
natural continuation, in this paper we compute the contributions of non-rigid
irreducible curves in the log Calabi-Yau setting and that of the union of two
rigid curves in general position. For the former, we construct and study a
moduli space of "logarithmic" 1-dimensional sheaves and compare the resulting
multiplicity with tropical multiplicity. For the latter, we explicitly describe
the components of the moduli space and work out the logarithmic deformation
theory in full, which we then compare with the deformation theory of the
analogous relative stable maps.
