We present an X-ray stacking analysis of ~75000 star-forming galaxies between 0.1<z<5.0 using the Chandra COSMOS-Legacy survey to study the X-ray emission of low-luminosity active galactic nuclei (AGN) and its connection to host galaxy properties. The stacks at z<0.9 have luminosity limits as low as 10^40^-10^41^erg/s, a regime in which X-ray binaries (XRBs) can dominate the X-ray emission. Comparing the measured luminosities to established XRB scaling relations, we find that the redshift evolution of the luminosity per star formation rate (SFR) of XRBs depends sensitively on the assumed obscuration and may be weaker than previously found. The XRB scaling relation based on stacks from the Chandra Deep Field South overestimates the XRB contribution to the COSMOS high specific SFR stacks, possibly due to a bias affecting the CDF-S stacks because of their small galaxy samples. After subtracting the estimated XRB contribution from the stacks, we find that most stacks at z>1.3 exhibit a significant X-ray excess indicating nuclear emission. The AGN emission is strongly correlated with stellar mass but does not exhibit an additional correlation with SFR. The hardness ratios of the high-redshift stacks indicate that the AGN are substantially obscured (N_H_~10^23^cm^-2^). These obscured AGN are not identified by IRAC color selection and have L_X_~10^41^-10^43^erg/s, consistent with accretion at an Eddington rate of ~10^-3^ onto 10^7^-10^8^M_{sun}_ black holes. Combining our results with other X-ray studies suggests that AGN obscuration depends on stellar mass and an additional variable, possibly the Eddington rate.