In random systems, the density of states of various linear problems, such as
phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits
an exponentially small Liftshitz tail at band edges. When the distribution of the
appropriate random variables (atomic masses, site energies, trap depths) has a
delta function at its lower (upper) bound, the Lifshitz singularities are pure
exponentials. We study in a quantitative way how these singularities are affected
by a universal logarithmic correction for continuous distributions starting with
a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders
in this logarithmic variable. For distributions starting with an essential
singularity, the exponent of the Lifshitz singularity itself is modified. These
results are obtained in the example of harmonic chains with random masses. It
is argued that analogous results hold in higher dimensions. Their implications
for other models, such as the long-time decay in trapping problems, are also
discussed.