  We use X-ray observations of three galaxies hosting LINERs (NGC 404, NGC
4736, NGC 4579) with Chandra to study their power sources. We find very diverse
properties within this small group: NGC 404 has an X-ray-faint nucleus with a
soft, thermal spectrum, NGC 4736 harbors a plethora of discrete X-ray sources
in and around its nucleus, and NGC 4579 has a dominant nuclear point source
embedded in a very extended, diffuse nebulosity. From their multi- wavelength
properties we conclude the following: the nucleus of NGC 404 is the site of a
weak, compact starburst, whose X-ray emission is due to gas heated by stellar
winds and supernovae, NGC 4736 is in a recent or aging starburst phase, where
the X-ray emission is dominated by a dense cluster of X-ray binaries, and NGC
4579 is powered by accretion onto a supermassive black hole. We detect 39
discrete sources in NGC 4736 and 21 in NGC 4579, most with L_X > 10^37 erg/s.
One source in the disk of NGC 4579 could be an ultraluminous X-ray binary with
L_X (2-10 keV) = 9x10^39 erg/s, but it could also be a background quasar. The
most luminous discrete sources have simple power-law spectra, which along with
their luminosities suggest that these are X-ray binaries accreting near or
above the Eddington rate for a neutron star. By comparing the luminosity
functions of discrete X-ray sources in these and other galaxies we find a
potential connection between the age of the stellar population and the slope of
the cumulative X-ray source luminosity function: galaxies with primarily old
stellar populations have steeper luminosity functions than starburst galaxies.
We suggest that this difference results from the contribution of high-mass
X-ray binaries from the young stellar population to the upper end of the
luminosity function.
