We study a model in which particles interact through a hard-core repulsion complemented by a short-ranged attractive potential of the kind found in colloidal suspensions. Combining theoretical and numerical work we locate the line of higher-order glass-transition singularities and its end point-named A4-on the fluid-glass line. Close to the A4 point, we detect logarithmic decay of density correlations and a sublinear power-law increase of the mean square displacement, for time intervals up to 4 orders of magnitude. We establish the presence of the A4 singularity by studying how the range of the potential affects the time window where anomalous dynamics is observed.