Motivated by quantum-mechanical corrections to the Newtonian potential, which can be translated into a -correction to the g00 component of the Schwarzschild metric, we construct a quantum-mechanically corrected metric assuming . We show how the Bekenstein black-hole entropy S receives its logarithmic contribution provided the quantum-mechanical corrections to the metric are negative. In this case the standard horizon at the Schwarzschild radius rS increases by small terms proportional to and a remnant of the order of Planck mass emerges.