  We study the bath correlation functions (BCFs) of open quantum systems
interacting with thermal baths, in case the spectral densities (SDs) exhibit
removable logarithmic singularities at low frequencies and are arbitrarily
shaped at higher frequencies. The singularities consist in arbitrarily positive
or negative powers of logarithmic functions, as additional factors for the
power laws of the Ohmic-like SDs. If the SD vanishes sufficiently fast at high
frequencies the short time behavior of the BCF is algebraic. The long time
behavior of the BCF exhibits a variety of relaxations that involve inverse
power laws and arbitrary powers of logarithmic forms. The imaginary part of the
BCF shows over long times regular dependence on the low frequency structure of
the SD, except for certain conditions where the ohmicity parameter takes odd
natural values. Same dependence holds for the real part of the BCF at
non-vanishing temperatures. At zero temperature the real part of the BCF
exhibits over long times the same regular relationship with the low frequency
structure of the SD, except for certain conditions involving even natural
values of the ohmicity parameter. The exceptional conditions provide
relaxations that are faster than those obtained via the regular dependence. In
this way, various long time relaxations of the BCF that are slower than
exponential decays and arbitrarily faster or slower than inverse power laws,
can be interpreted in terms of removable logarithmic singularities in the low
frequency structure SD of an open quantum system.
