Plotting a function in log-scale is a very common way to "compress" a function that grows too fast. What I want to do is to use a different function to compress a plot by. For example, I would like to scale the Y-Axis by a function I call the "bilateral logarithm" which is given by
BLog[x_] = Sign[x] Log2[Abs[x] + 1];
This function is nice because it is asymptotic to -Log2(-x) as x approaches -inf, Log2(x) as x approaches +inf, and BLog(-1)=-1, BLog(0)=0, BLog(1)=1. And all of this always holds for nested Blog(Blog(...)).
In summary, for a log-scale plot, the distances on the y-axis are proportional to Log(y), and I want to make it proportional to BLog(y). How do I make it happen?
edit: the inverse of this function is
Sign[x] (2^Abs[x] - 1)