How to calculate galaxy bolometric luminosity?
The bolometric correction is the difference between a bolometric magnitude and the magnitude in some band.
$$BC = M_{\rm bol} - M_{5100} = -2.5\log\left(\frac{L_{\rm bol}}{L_{5100}}\right)
$$
$$\log L_{\rm bol} = \log L_{5100} - 0.4BC \ ,$$
$$ \log L_{\rm bol} = \log L_{5100} +0.4\log L_{5100} - 21.2  \ ,$$
$$ \log L_{\rm bol} = 1.4(1.4 \log L_x -16.8) - 21.2\ , $$
$$ \log L_{\rm bol} = 1.96\log L_x  -44.72\ .$$