  It has been known for many years that the leading correction to the black
hole entropy is a logarithmic term, which is universal and closely related to
conformal anomaly. A fully consistent analysis of this issue has to take
quantum backreactions to the black hole geometry into account. However, it was
always unclear how to naturally derive the modified black hole metric
especially from an effective action, because the problem refers to the elusive
non-locality of quantum gravity. In this paper, we show that this problem can
be resolved within an effective field theory (EFT) framework of quantum
gravity. Our work suggests that the EFT approach provides a powerful and
self-consistent tool for studying the quantum gravitational corrections to
black hole geometries and thermodynamics.
