Materials tuned to a quantum critical point display universal scaling properties as a function of temperature T and frequency ω . A long-standing puzzle regarding cuprate superconductors has been the observed power-law dependence of optical conductivity with an exponent smaller than one, in contrast to T -linear dependence of the resistivity and ω -linear dependence of the optical scattering rate. Here, we present and analyze resistivity and optical conductivity of La _{2− x} Sr _x CuO ₄ with x = 0.24. We demonstrate ℏ ω / k _B T scaling of the optical data over a wide range of frequency and temperature, T -linear resistivity, and optical effective mass proportional to $$\sim \ln T$$ ~ ln T corroborating previous specific heat experiments. We show that a T , ω -linear scaling Ansatz for the inelastic scattering rate leads to a unified theoretical description of the experimental data, including the power-law of the optical conductivity. This theoretical framework provides new opportunities for describing the unique properties of quantum critical matter.