  We introduce a new concept of logarithmic topological recursion that provides
a patch to topological recursion in the presence of logarithmic singularities
and prove that this new definition satisfies the universal $x-y$ swap relation.
This result provides a vast generalization and a proof of a very recent
conjecture of Hock. It also uniformly explains (and conceptually rectifies) an
approach to the formulas for the $n$-point functions proposed by Hock.
