  Within the mode-coupling theory for ideal glass transitions, an analysis for
the correlation functions of glass-forming systems for states near higher-order
glass-transition singularities is presented. It is shown that the solutions of
the equations of motion can be asymptotically expanded in polynomials of the
logarithm of time t. In leading order, an ln(t)-law is obtained, and the
leading corrections are given by a fourth-order polynomial. The correlators
interpolate between three scenarios. First, there are planes in parameter space
where the dominant corrections to the ln(t)-law vanish, so that the logarithmic
decay governs the structural relaxation process. Second, the dynamics due to
the higher-order singularity can describe the initial and intermediate part of
the alpha-process thereby reducing the range of validity of von Schweidler's
law and leading to strong alpha-relaxation stretching. Third, the ln(t)-law can
replace the critical decay law of the beta-process leading to a particularly
large crossover interval between the end of the transient and the beginning of
the alpha-process. This may lead to susceptibility spectra below the band of
microscopic excitations exhibiting two peaks. Typical results of the theory are
demonstrated for models dealing with one and two correlation functions.
