In this work, a new single formula is proposed for the law of the wall, which is dubbed as the LOG-EXP (logarithmic-exponential) formula, for predicting the mean velocity profile in different regions near the wall. When deriving the LOG-EXP formula, a correction term instead of a damping function is introduced to model the difference between the mixing-length eddy viscosity model and the actual Reynolds shear stress for the near-wall effect. The formula is tested using the direct numerical simulation (DNS) data of turbulent channel flows, the experimental data of the boundary layer flow over a flat plate, and the circular pipe flow for different Reynolds numbers. The formula is then employed as the wall model in wall-modeled large-eddy simulation (WMLES) and tested using the turbulent channel flow at three different Reynolds numbers. An overall good performance of the LOG-EXP formula is observed for all the considered cases. When refining the grid, no significant improvements are observed for the Reynolds stresses, as there are several factors, e.g., the grid employed, and the subgrid-scale model, affecting the predictive capability of WMLES. Further work is needed to quantify their effects and improve the WMLES predictions on a fine grid but not fine enough for a wall-resolved large-eddy simulation. (c) 2022 Elsevier Masson SAS. All rights reserved.