  In this paper we consider the logarithmic Schr\"{o}dinger equation on a star
graph. By using a compactness method, we construct a unique global solution of
the associated Cauchy problem in a suitable functional framework. Then we show
the existence of several families of standing waves. We also prove the
existence of ground states as minimizers of the action on the Nehari manifold.
Finally, we show that the ground states are orbitally stable via a variational
approach.
