We study here higher-order two-loop and three-loop logarithmic contributions to the Lamb shift in light hydrogenlike atoms. We have found the leading logarithmic contributions in order ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{7}m$ and ${\ensuremath{\alpha}}^{3}{(Z\ensuremath{\alpha})}^{6}m$ for the arbitrary states. Those terms are double logarithmic for the $ns$ states and single logarithmic for the $np$ states and vanish for the states with $l\ensuremath{\ge}2$. We have also obtained the leading (single-logarithmic) contribution to the specially normalized difference for the Lamb shift of the $ns$ states ${\mathrm{\ensuremath{\Delta}}}_{L}(n)={E}_{L}(1s)\ensuremath{-}{n}^{3}{E}_{L}(ns)$, which is important for a combined evaluation of the overall set of the experimental data available on various transitions in hydrogen and deuterium.