  We have conducted a spectroscopic survey to find faint quasars (-26.0 <
M_{1450} < -22.0) at redshifts z=3.8-5.2 in order to measure the faint end of
the quasar luminosity function at these early times. Using available optical
imaging data from portions of the NOAO Deep Wide-Field Survey and the Deep Lens
Survey, we have color-selected quasar candidates in a total area of 3.76 deg^2.
Thirty candidates have R <= 23 mags. We conducted spectroscopic followup for 28
of our candidates and found 23 QSOs, 21 of which are reported here for the
first time, in the 3.74 < z <5.06 redshift range. We estimate our survey
completeness through detailed Monte Carlo simulations and derive the first
measurement of the density of quasars in this magnitude and redshift interval.
We find that the binned luminosity function is somewhat affected by the
K-correction used to compute the rest-frame absolute magnitude at 1450A.
Considering only our R <= 23 sample, the best-fit single power-law (Phi \propto
L^beta) gives a faint-end slope beta = -1.6+/-0.2. If we consider our larger,
but highly incomplete sample going one magnitude fainter, we measure a steeper
faint-end slope -2 < beta < -2.5. In all cases, we consistently find faint-end
slopes that are steeper than expected based on measurements at z ~ 3. We
combine our sample with bright quasars from the Sloan Digital Sky Survey to
derive parameters for a double-power-law luminosity function. Our best fit
finds a bright-end slope, alpha = -2.4+/-0.2, and faint-end slope, beta =
-2.3+/-0.2, without a well-constrained break luminosity. This is effectively a
single power-law, with beta = -2.7+/-0.1. We use these results to place limits
on the amount of ultraviolet radiation produced by quasars and find that
quasars are able to ionize the intergalactic medium at these redshifts.
