Let be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize under the assumption that there exists a commutative algebra in with certain properties: Let be the category of local -modules in and the category of -modules in , which are in our set-up usually much simpler categories than . Then we can characterize as a relative Drinfeld center and as representations of a certain Hopf algebra inside . In particular this allows us to reduce braided tensor equivalences to the knowledge of abelian equivalences, e.g. if we already know that is abelian equivalent to the category of modules of some quantum group or some generalization thereof, and if is braided equivalent to a category of graded vector spaces, and if has a certain form, then we already obtain a braided tensor equivalence between and . A main application of our theory is to prove logarithmic Kazhdan-Lusztig correspondences, that is, equivalences of braided tensor categories of representations of vertex algebras and of quantum groups. Here, the algebra and the corresponding category are a-priori given by a free-field realization of the vertex algebra and by a Nichols algebra. We illustrate this in those examples where the representation theory of the vertex algebra is well enough understood. In particular we prove the conjectured correspondences between the singlet vertex algebra and the unrolled small quantum groups of at -th root of unity. Another new example is the Kazhdan-Lusztig correspondence for .