The logarithmic torsion homology growth of a group is the limit superior of the logarithm of the size of the homology torsion of finite index subgroups, normalised by the index. This invariant encodes geometric, dynamical, and number-theoretic aspects of groups (and spaces). In general, exact computations tend to be difficult and despite of recent progress in various directions many fundamental questions are wide open. In this talk, I will outline techniques to show vanishing results for logarithmic torsion homology growth based on measured group theory; in particular, I will illustrate these techniques in the well-understood case of amenable groups.