  We present a measurement of the Type I quasar luminosity function at z=5
using a large sample of spectroscopically confirmed quasars selected from
optical imaging data. We measure the bright end (M_1450<-26) with Sloan Digital
Sky Survey (SDSS) data covering ~6000 deg^2, then extend to lower luminosities
(M_1450<-24) with newly discovered, faint z~5 quasars selected from 235 deg^2
of deep, coadded imaging in the SDSS Stripe 82 region (the celestial equator in
the Southern Galactic Cap). The faint sample includes 14 quasars with spectra
obtained as ancillary science targets in the SDSS-III Baryon Oscillation
Spectroscopic Survey (BOSS), and 59 quasars observed at the MMT and Magellan
telescopes. We construct a well-defined sample of 4.7<z<5.1 quasars that is
highly complete, with 73 spectroscopic identifications out of 92 candidates.
Our color selection method is also highly efficient: of the 73 spectra
obtained, 71 are high redshift quasars. These observations reach below the
break in the luminosity function (M_1450* ~ -27). The bright end slope is steep
(beta <~ -4), with a constraint of beta < -3.1 at 95% confidence. The break
luminosity appears to evolve strongly at high redshift, providing an
explanation for the flattening of the bright end slope reported previously. We
find a factor of ~2 greater decrease in the number density of luminous quasars
(M_1450<-26) from z=5 to z=6 than from z=4 to z=5, suggesting a more rapid
decline in quasar activity at high redshift than found in previous surveys. Our
model for the quasar luminosity function predicts that quasars generate ~30% of
the ionizing photons required to keep the universe ionized at z=5.
