We study the global SF law - the relation between gas and SFRs in a sample of 181 local galaxies with L_IR spanning almost five orders of magnitude, which includes 115 normal galaxies and 66 (U)LIRGs. We derive their atomic, molecular gas and dense molecular gas masses using newly available HI, CO and HCN data from the literature, and SFRs are determined both from total IR and 1.4 GHz radio continuum (RC) luminosities. In order to derive the disk-averaged surface densities of gas and SFRs, we have used high-resolution RC observations to measure the radio sizes for all galaxies. We find that dense molecular gas (as traced by HCN) has the tightest correlation with that of SFRs, and is linear in (N=1.01 +/- 0.02) across the full galaxy sample. The correlation between densities of molecular gas (traced by CO) and SFRs is sensitive to the adopted value of the alpha_CO used to infer molecular gas masses from CO luminosities. For a fixed value of alpha_CO, a slope of 1.14+/-0.02 is found. If instead we adopt values of 4.6 and 0.8 for disk galaxies and (U)LIRGs, respectively, we find the two distinct relations. If applying a continuously varying alpha_CO to our sample, we recover a single relation with slope of 1.60+/-0.03. The SFRs is a steeper function of total gas than that of molecular gas, and is tighter among low-luminosity galaxies. We find no correlation between SFRs and atomic gas.
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