Barenblatt [J. Fluid Mech.248, 513 (1993)] proposed a Reynolds number(R)-dependent power law for the mean velocity near the wall in fully developed turbulent shear flows. In an appropriate region, asR→∞, the envelope of the velocity profiles reduces to a logarithmic law with an additive constant that is significantly larger than that of the universal logarithmic law. This discrepancy is removed upon including a higher-order term in the power-law exponent. Using the friction (instead of average) velocity in the definition ofRdestroys the invariance of the nonvanishing higher-order terms of the power law asR→∞. ©1998 American Institute of Physics.