  In this paper, we develop the theory of relative log convergent cohomology.
We prove the coherence of relative log convergent cohomology in certain case by
using the comparison theorem between relative log convergent cohomlogy and
relative log crystalline cohomology, and we relates relative log convergent
cohomology to relative rigid cohomology to show the validity of Berthelot's
conjecture on the coherence and the overconvergence of relative rigid
cohomology for proper smooth families when they admit nice proper log smooth
compactification to which the coefficient extends logarithmically.
