For some reason you're getting a lot of wrong or just bad replies. But the answer to your question is yes both sRGB/gamma2.2 & log are non-linear, but almost in the opposite direction. gamma2.2 is exponential not logarithmic. As in, it's spending all its bits on the lower half of the brightness range, whereas log is actually spending more bits in the highlights.
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.)
That's neither logarithmic nor what log cameras capture. See the link posted by the sibling comment[1] for the actual curves.
If you see an S-curve, that's usually what you will try to map the captured images too because it allows for increased detail in both shadows and highlights, while allowing a natural dynamic range in the middle areas. Log capturing allows you to have a much higher dynamic range (with a given number of bits), and thus more easily map to the S-curve that you want.
I think that s-curve is the target (i.e. the overall end-to-end system gamma, combining the encoding and decoding transfer functions). If it's linear to reproduce the source, but for various reasons [1] sometimes it's preferable to have a gradual roll-of
It actually looks more like HLG in this way.
https://www.artstation.com/blogs/tiberius-viris/3ZBO/color-s... has some plots of the curves to compare visually