For nearly a century, the universal logarithmic law of the mean velocity profile has been a mainstay of turbulent fluid mechanics and its teaching. Yet many experiments and numerical simulations are not fit exceedingly well by it, and the question whether the logarithmic law is indeed universal keeps turning up in discussion and in writing. Large experiments have been set up in various parts of the world to confirm or deny the logarithmic law and accurately estimate von Kármán's constant, the coefficient that governs it. Here, we show that the discrepancy among flows in different (circular or plane) geometries can be ascribed to the effect of the pressure gradient. When this effect is accounted for in the form of a higher-order perturbation, universal agreement emerges beyond doubt and a satisfactorily simple formulation is established.