Time-average velocity distribution in steady and uniform channel flows is important for fundamental research and practical application as it is always three-dimensional (3D), regardless of channel geometry. However, its determination has predominantly been carried out by using complex numerical software, even for the simplest geometry such as rectangular channels. The log-law was developed initially for circular pipe flows, where a single shear velocity is used to normalize the velocity (u+) and its distance (y+). Tracy and Lester found that the performance of the log-law can be extended to express velocity profiles in rectangular channels when the global shear velocities (gRS)0.5 and (ghS)0.5 are used to normalize the measured velocity u and its distance y. This study extends this discovery from the channel central line to the corner regions, and its general form of log-law was found to be valid even in trapezoidal or triangular open channels or closed ducts. This modified log-law can produce good agreement with the measured velocity with an average error of less than 5%. Therefore, this study provides a simple and reliable tool for engineers and researchers to estimate the velocity contours in straight and smooth channel flows.