We present results from the MOSFIRE Deep Evolution Field (MOSDEF) survey on the identification, selection biases, and host galaxy properties of 55 X-ray, IR and optically-selected active galactic nuclei (AGN) at $1.4 < z < 3.8$. We obtain rest-frame optical spectra of galaxies and AGN and use the BPT diagram to identify optical AGN. We examine the uniqueness and overlap of the AGN identified at different wavelengths. There is a strong bias against identifying AGN at any wavelength in low mass galaxies, and an additional bias against identifying IR AGN in the most massive galaxies. AGN hosts span a wide range of star formation rate (SFR), similar to inactive galaxies once stellar mass selection effects are accounted for. However, we find (at $sim 2-3sigma$ significance) that IR AGN are in less dusty galaxies with relatively higher SFR and optical AGN in dusty galaxies with relatively lower SFR. X-ray AGN selection does not display a bias with host galaxy SFR. These results are consistent with those from larger studies at lower redshifts. Within star-forming galaxies, once selection biases are accounted for, we find AGN in galaxies with similar physical properties as inactive galaxies, with no evidence for AGN activity in particular types of galaxies. This is consistent with AGN being fueled stochastically in any star-forming host galaxy. We do not detect a significant correlation between SFR and AGN luminosity for individual AGN hosts, which may indicate the timescale difference between the growth of galaxies and their supermassive black holes.
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