We present new Chandra observations that complete a sample of seventeen (17) luminous infrared galaxies (LIRGs) with D < 60 Mpc and low Galactic column densities of N_H < 5 X 10^20 cm^-2. The LIRGs in our sample have total infrared (8-1000um) luminosities in the range of L_IR ~ (1-8) X 10^11 L_sol. The high-resolution imaging and X-ray spectral information from our Chandra observations allow us to measure separately X-ray contributions from active galactic nuclei (AGNs) and normal galaxy processes (e.g., X-ray binaries and hot gas). We utilized total infrared plus UV luminosities to estimate star-formation rates (SFRs) and K-band luminosities and optical colors to estimate stellar masses (M*) for the sample. Under the assumption that the galaxy-wide 2-10 keV luminosity (LX) traces the combined emission from high mass X-ray binaries (HMXBs) and low mass X-ray binaries (LMXBs), and that the power output from these components are linearly correlated with SFR and M*, respectively, we constrain the relation LX = alpha M* + beta SFR. To achieve this, we construct a Chandra-based data set composed of our new LIRG sample combined with additional samples of less actively star-forming normal galaxies and more powerful LIRGs and ultraluminous infrared galaxies (ULIRGs) from the literature. Using these data, we measure best-fit values of alpha = (9.05 +/- 0.37) X 10^28 ergs s^-1 Msol^-1 and beta = (1.62 +/- 0.22) X 10^39 ergs s^-1 (Msol yr^-1)^-1. This scaling provides a more physically meaningful estimate of LX, with ~0.1-0.2 dex less scatter, than a direct linear scaling with SFR (abridged).
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