The integral of  thus yields the logarithm of the ratio of the upper and lower cut-offs. This number is known as the Coulomb logarithm and is designated by either  or . It is the factor by which small-angle collisions are more effective than large-angle collisions. The Coulomb logarithm was introduced independently by Lev Landau in 1936 and Subrahmanyan Chandrasekhar in 1943. For many plasmas of interest it takes on values between  and . (For convenient formulas, see pages 34 and 35 of the [https://www.nrl.navy.mil/ppd/content/nrl-plasma-formulary NRL Plasma formulary].) The limits of the impact parameter integral are not sharp, but are uncertain by factors on the order of unity, leading to theoretical uncertainties on the order of . For this reason it is often justified to simply take the convenient choice . The analysis here yields the scalings and orders of magnitude.