  In this paper we obtain logarithmic Hardy and Rellich inequalities on general
Lie groups. In the case of graded groups, we also show their refinements using
the homogeneous Sobolev norms. In fact, we derive a family of weighted
logarithmic Hardy-Rellich inequalities, for which logarithmic Hardy and Rellich
inequalities are special cases. As a consequence of these inequalities, we also
derive a Gross type logarithmic Hardy inequality on general stratified groups.
An interesting feature of such estimate is that we consider the measure which
is Gaussian only on the first stratum of the group. Such choice of the measure
is natural in view of the known Gross type logarithmic Sobolev inequalities on
stratified groups. The obtained results are new already in the setting of the
Euclidean space $\mathbb R^n.$ Finally, we also present a simple argument for
getting a logarithmic Poincar\'e inequality, as well as the logarithmic Hardy
inequality for the fractional $p$-sub-Laplacian on homogeneous groups.
