  We report HCN J=4-3, HCO+ J=4-3, and CS J=7-6 observations in 20 nearby
star-forming galaxies with the Acatama Pathfinder EXperiment 12-m telescope.
Combined with 4 HCN, 3 HCO+, and 4 CS detections in literature, we probe the
empirical link between the luminosity of molecular gas (L_gas) and that of
infrared emission (L_IR), up to the highest gas densities (10^6 - 10^8 cm-3)
that have been probed so far. For nearby galaxies with large radii, we measure
the IR luminosity within the submm beam-size (14"-18") to match the molecular
emission. We find linear slopes for L_CS76-L_IR and L_HCN43-L_IR, and a
slightly super-linear slope for L_HCO+43-L_IR. The correlation of L_CS76-L_IR
even extends over eight orders of luminosity magnitude down to Galactic dense
cores, with a fit of log(L_IR)=1.00(\pm 0.01) \times log(L_CS76) + 4.03(\pm
0.04). Such linear correlations appear to hold for all densities >10^4 cm-3,
and indicate that star formation rate is not related to free-fall time scale
for dense molecular gas.
