  We investigate the quasi-bound states of a Coulomb impurity in graphene in
the presence of a magnetic field. \ These states exhibit the dramatic and
rather rare property of discrete scale invariance when the Coulomb potential is
supercritical. \ We show using both Wentzel-Kramers-Brillouin (WKB)
approximation and numerical studies that the supercritical states are converted
to subcritical states as the field is increased. \ The local density of states
is calculated and it shows direct signatures of discrete scale invariance. \ In
a magnetic field, these signatures are gradually destroyed in a systematic way.
\ Hence the effect that we propose can be detected via scanning tunneling
microscope experiments. \ The range of magnetic field and energy resolution
required are compatible with existing experimental setups. \ These experiments
can be performed in a single sample by changing the field; they do not involve
changing the nuclear charge.
