  We introduce the notion of a holonomic D-module on a smooth (idealized)
logarithmic scheme and show that Verdier duality can be extended to this
context. In contrast to the classical case, the pushforward of a holonomic
module along an open immersion is in general not holonomic. We introduce a
"perverse" t-structure on the category of coherent logarithmic D-modules which
makes the dualizing functor t-exact on holonomic modules. This allows us to
transfer some of the formalism from the classical setting and in particular
show that every holonomic module on an open subscheme can be extended to a
holonomic module on the whole space. Conversely this t-exactness characterizes
holonomic modules among all coherent logarithmic D-modules. We also introduce
logarithmic versions of the Gabber and Kashiwara-Malgrange filtrations.
