Josh,

I think you want to take a brief foray into psychophysics.
The function is not an s-curve.  Take a look at Steven's
Power Law.  These links should get you started:

http://www.psychology.adelaide.edu.au/psychophysics/Library/
FD99/Contribute/Stevensetal.pdf

http://nersp.nerdc.ufl.edu/~ufruss/Research%20Design/magnitu
de%20scaling.htm

Best regards,
Michael

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