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The single-layer Mott insulator strontium iridate Sr$_2$IrO$_4$ was studied using a scanning tunneling microscope. This measurement technique is unique due to the transport properties of this Mott insulator allowing tunneling measurements to be performed, even at cryogenic temperatures. We obtained high-resolution images of the sample surface and the differential tunneling conductance at different cryogenic temperatures. The differential conductance is a direct measurement of the local electronic density of states which provided an insulating gap consistent with optical conductivity, angle resolved photoemission spectroscopy and resonant inelastic x-ray scattering (RIXS) experiments. The observed widths of these features is broader than predicted by the Slater approximation and narrower than predicted by dynamical mean field theory. Additionally, the observed density of states due to magnetic fluctuations is found in the derivative of the differential conductance and is consistent with results from Raman scattering and RIXS. At low temperatures, additional low-energy features were observed, suggesting a change in the dispersion of the collective magnetic excitations, which is consistent with the magnetic susceptibility., Comment: 5 pages, 4 figures, submitted to Phys. Rev. B