We present a detailed analysis of the relation between infrared luminosity and molecular line luminosity, for a variety of molecular transitions, using a sample of 34 nearby galaxies spanning a broad range of infrared luminosities (10^{10} < L_{IR} < 10^{12.5} L_sun). We show that the power-law index of the relation is sensitive to the critical density of the molecular gas tracer used, and that the dominant driver in observed molecular line ratios in galaxies is the gas density. As most nearby ultraluminous infrared galaxies (ULIRGs) exhibit strong signatures of active galactic nuclei (AGN) in their center, we revisit previous claims questioning the reliability of HCN as a probe of the dense gas responsible for star formation in the presence of AGN. We find that the enhanced HCN(1-0)/CO(1-0) luminosity ratio observed in ULIRGs can be successfully reproduced using numerical models with fixed chemical abundances and without AGN-induced chemistry effects. We extend this analysis to a total of ten molecular line ratios by combining the following transitions: CO(1-0), HCO+(1-0), HCO+(3-2), HCN(1-0), and HCN(3-2). Our results suggest that AGNs reside in systems with higher dense gas fraction, and that chemistry or other effects associated with their hard radiation field may not dominate (NGC 1068 is one exception). Galaxy merger could be the underlying cause of increased dense gas fraction and the evolutionary stage of such mergers may be another determinant of the HCN/CO luminosity ratio.
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