  We propose a logarithmic enhancement of the Gromov-Witten/Donaldson-Thomas
correspondence, with descendants, and study the behaviour of the correspondence
under simple normal crossings degenerations. The formulation of the logarithmic
correspondence requires a matching of tangency conditions and relative
insertions. This is achieved via a version of the Nakajima basis for the
cohomology of the Hilbert schemes of points on logarithmic surfaces.
  We then establish a strong form of the degeneration formula in logarithmic DT
theory - the numerical DT invariants of the general fiber of a degeneration are
determined by the numerical DT invariants attached to strata of the special
fiber. The GW version of this result, which we also prove here, is a
strengthening of the currently known formulas. A key role is played by a
certain exotic class of insertions, introduced here, and can be thought of as
non-local incidence conditions coupled across multiple boundary strata of the
target geometry.
  Finally, we prove compatiblity of the new logarithmic GW/DT correspondence
with degenerations, and in particular, knowledge of the conjecture on the
strata of the special fiber of a degeneration implies it on the general fiber.
Several examples are included to illustrate the nature and utility of the
formula.
