  In this paper we will demonstrate possible existence of more exotic forms of
interaction taking into account, that dark energy can be parametrized as a
varying polytropic uid suggested recently by the first author. On going
research in both directions, i.e. dark energy and non-gravitational
interaction, opens very interesting perspectives for solving the problems which
are related to the accelerated expansion of the Universe. The fact that
available observational data allows to parametrize the dark side of our
Universe in form of interacting dark uids is not less surprising, than the
accelerated expansion itself. Therefore, the consideration of new forms of
non-gravitational interactions in modern cosmology still is one of the actual
topics and requires deep and systematic research. This fact motivates us to
consider various new forms of non-gravitational logarithmic interactions and
study their cosmological consequences. Appropriate classification of the models
is presented in the light of actively discussed Om analysis and the result for
the Hubble parameter measured at z = 2:34 in BOSS experiment.
