The Kondo resonance of Co adatoms on the Cu(100) and Cu(111) surfaces has been studied by scanning tunneling spectroscopy. We demonstrate the scaling of the Kondo temperature T(K) with the host electron density at the magnetic impurity. The quantitative analysis of the tunneling spectra reveals that the Kondo resonance is dominated by the Cu bulk electrons. While at the Cu(100) surface both tunneling into the hybridized localized state and into the substrate conduction band contribute to the Kondo resonance, the latter channel is found to be dominant for Cu(111).