Inner-amenability is a weak form of amenability, which is satisfied e.g. by products where one factor is infinite amenable. Some properties of amenable groups extend to inner-amenable groups, e.g. the vanishing of the first $\ell^2$-Betti number. In this talk, we will treat logarithmic torsion homology growth. One tool for showing vanishing of this invariant is the cheap rebuilding property of Abért, Bergeron, Frączyk and Gaboriau. Certain inner-amenable groups have this property in degree one, thus extending vanishing results that were already known for amenable groups.