  We study log D-modules on smooth log pairs and construct a comparison theorem
of log de Rham complexes. The proof uses Sabbah's generalized b-functions. As
applications, we deduce a log index theorem and a Riemann-Roch type formula for
perverse sheaves on smooth quasi-projective varieties. The log index theorem
naturally generalizes the Dubson-Kashiwara index theorem on smooth projective
varieties.
