The location and the manner of union of the H0 and HR acidity functions with −log [Formula: see text] have been determined for aqueous HClO4, HCl, and H2SO4 (H0 only) through particularly careful and extensive indicator measurements in dilute and moderately concentrated solutions of these acids. These data were also used to evaluate and compare a number of different ways of extrapolating measurements made in concentrated acids down to dilute solution; over the limited range of acidity investigated, the traditional acidity function method, as commonly applied in the absence of an appropriate acidity function (i.e. in the form of mH0), was found to be the least satisfactory of these extrapolative methods, and the Cox–Yates technique was found to be the best.