We use deep Chandra imaging to measure the distribution of X-ray luminosities (L_X) for samples of star-forming galaxies as a function of stellar mass and redshift, using a Bayesian method to push below the nominal X-ray detection limits. Our luminosity distributions all show narrow peaks at L_X < 10^{42} erg/s that we associate with star formation, as opposed to AGN that are traced by a broad tail to higher L_X. Tracking the luminosity of these peaks as a function of stellar mass reveals an X-ray main sequence with a constant slope ~0.63 +/- 0.03 over 8.5 < log M*/Msun < 11.5 and 0.1 < z < 4, with a normalization that increases with redshift as (1+z)^{3.79+/-0.12}. We also compare the peak X-ray luminosities with UV-to-IR tracers of star formation rates (SFRs) to calibrate the scaling between L_X and SFR. We find that L_X propto SFR^{0.83} x (1+z)^{1.3}, where the redshift evolution and non-linearity likely reflect changes in high-mass X-ray binary populations of star-forming galaxies. Using galaxies with a broader range of SFR, we also constrain a stellar-mass-dependent contribution to L_X, likely related to low-mass X-ray binaries. Using this calibration, we convert our X-ray main sequence to SFRs and measure a star-forming main sequence with a constant slope ~0.76+/-0.06 and a normalization that evolves with redshift as (1+z)^{2.95+/-0.33}. Based on the X-ray emission, there is no evidence for a break in the main sequence at high stellar masses, although we cannot rule out a turnover given the uncertainties in the scaling of L_X to SFR.
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