  In the first part we deepen the six-functor theory of (holonomic) logarithmic
D-modules, in particular with respect to duality and pushforward along
projective morphisms. Then, inspired by work of Ogus, we define a logarithmic
analogue of the de Rham functor, sending logarithmic D-modules to certain
graded sheaves on the so-called Kato-Nakayama space. For holonomic modules we
show that the associated sheaves have finitely generated stalks and that the de
Rham functor intertwines duality for D-modules with a version of
Poincar\'e-Verdier duality on the Kato-Nakayama space. Finally, we explain how
the grading on the Kato-Nakayama space is related to the classical
Kashiwara-Malgrange V-filtration for holonomic D-modules.
