f(h)=eh
h(x)=ex
f(x)=eex
That's a double exponential instead of a double-log S curve, but the same transformation applies there too.
Nice. Only, that's not the right composition for the scenario you described, where the numerator exponentially increases in value and the denominator exponentially decreases in value. You composition actually makes the rate at which BTC increases dependent on the rate at which USD decreases. Was that what you had in mind?
I don't know if you have the patience reading all this in the following page, but the far bottom graphs might give you an idea what you are looking for. I've expressed my opinion on this earlier in this thread but among my enormous urge for trolling this might as well got forgotten.