Does there exist a fractional logarithm operator?
Something like this:
$$L_0(x) = x$$ $$L_1(x) = log(x)$$ $$L_{0.5}(x) = ???$$
This is the motivating situation: consider perception of sound, which is logarithmic. So a sequence of sounds with amplitude 1, 10, 100 is perceived to be linearly increasing in volume.
If we are plotting a spectrum, we wouldn't plot amplitude against frequency. This wouldn't look right as per our perception. A little bump next to a mountain, it would look as though that should be discarded. But that bump is audible. Taking logarithms, that bump now becomes visually significant.
If I could take a $L_{1.5}(\cdot)$ fractional logarithm, it would increase the prominence of these minor features further.
Maybe I can consider an analogy with powers. Between $x$ and $x^2$ we can have $f(x)$ such that $f_2(x) = f(f(x)) = x^2$.
hmm no I am stuck!
PS if anyone can suggest a mapping that accomplishes what I'm after, please do suggest, even if it doesn't directly answer the original question!