  We investigate logarithmic corrections to the entropy of supersymmetric,
rotating, asymptotically AdS$_5$ black holes and black strings. Within the
framework of the AdS/CFT correspondence, the entropy of these black objects is
determined, on the field theory side, by the superconformal index and the
refined topologically twisted index of $\mathcal{N}=4$ supersymmetric
Yang-Mills theory, respectively. We read off the logarithmic correction from
those field-theoretic partition functions. On the gravity side, we take the
near-horizon limit and apply the Kerr/CFT correspondence whose associated
charged Cardy formula describes the degeneracy of states at subleading order
and determines the logarithmic correction to the entropy. We find perfect
agreement between these two approaches. Our results provide a window into
precision microstate counting and demonstrate the efficacy of low-energy,
symmetry-based approaches such as the Kerr/CFT correspondence for
asymptotically AdS black objects under certain conditions.
