  An insightful argument for a linear relation between the entropy and the area
of a black hole was given by Bekenstein using only the energy-momentum
dispersion relation, the uncertainty principle, and some properties of
classical black holes. Recent analyses within String Theory and Loop Quantum
Gravity describe black-hole entropy in terms of a dominant contribution, which
indeed depends linearly on the area, and a leading log-area correction. We
argue that, by reversing the Bekenstein argument, the log-area correction can
provide insight on the energy-momentum dispersion relation and the uncertainty
principle of a quantum-gravity theory. As examples we consider the
energy-momentum dispersion relations that recently emerged in the Loop Quantum
Gravity literature and the Generalized Uncertainty Principle that is expected
to hold in String Theory.
