  In a beautiful paper Deligne and Illusie proved the degeneration of the
Hodge-to-de Rham spectral sequence using positive characteristic methods. In a
recent paper Arinkin, C\u{a}ld\u{a}raru and the author of this paper gave a
geometric interpretation of the problem of Deligne-Illusie showing that the
triviality of a certain line bundle on a derived scheme implies the the
Deligne-Illusie result. In the present paper we generalize these ideas to
logarithmic schemes and using the theory of twisted derived intersection of
logarithmic schemes we obtain the Hodge theorem for the logarithmic de Rham
complex.
