Black holes (BH's) in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly SU(2) invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with punctures. Remarkably, when considering an ensemble of fixed horizon area a(H), the counting can be mapped to simply counting the number of SU(2) intertwiners compatible with the spins labeling the punctures. The resulting BH entropy is proportional to a(H) with logarithmic corrections ΔS=-3/2 loga(H). Our treatment from first principles settles previous controversies concerning the counting of states.