The difficulty of determining the slope of the famed logarithmic law in the mean velocity profile of wall-bounded turbulent flows, the inverse of the K & aacute;rm & aacute;n "constant" kappa, from direct numerical simulations (DNS) is discussed for channel flow. Unusual approaches as well as the analysis of the standard log-indicator function are considered and analyzed, including the identification of the higher order linear overlap term. This leads to the conclusion that a definitive determination of the channel flow kappa from DNS with an uncertainty of, say, 2%-3% will require the residue of the mean streamwise momentum equation in the simulations to be reduced by at least an order of magnitude. (c) 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).