I think you're mixing up OOTFs and EOTFs here. sRGB or HLG can refer to either the stored gamma, but more often means the EOTF "reversed" gamma that is used to display an image. When we refer to "log", this is almost always means a camera gamma - an OOTF. So the reason it's "in the opposite direction" is that it's designed to efficiently utilize bits for storing image data, whereas the EOTF is designed to reverse this storage gamma for display purposes.
As you can see from the graph in [1], Sony's S-Log does indeed allocate more bits to dark areas than bright areas. (Though the shape of the curve becomes more complicated if you take into account the non-linear behavior of light in human vision.)
I think you're mixing up OOTFs and EOTFs here. sRGB or HLG can refer to either the stored gamma, but more often means the EOTF "reversed" gamma that is used to display an image. When we refer to "log", this is almost always means a camera gamma - an OOTF. So the reason it's "in the opposite direction" is that it's designed to efficiently utilize bits for storing image data, whereas the EOTF is designed to reverse this storage gamma for display purposes.
As you can see from the graph in [1], Sony's S-Log does indeed allocate more bits to dark areas than bright areas. (Though the shape of the curve becomes more complicated if you take into account the non-linear behavior of light in human vision.)
[1] https://www.enriquepacheco.com/blog/s-log-tutorial