  In this paper we consider the nonlinear fractional logarithmic
Schr\"{o}dinger equation. By using a compactness method, we construct a unique
global solution of the associated Cauchy problem in a suitable functional
framework. We also prove the existence of ground states as minimizers of the
action on the Nehari manifold. Finally, we prove that the set of minimizers is
a stable set for the initial value problem, that is, a solution whose initial
data is near the set will remain near it for all time.
