I have some functions of $x$, in the form of $d\sqrt{x}$ or $d\log(x)$ where d is known. I would like to rewrite (approximate is fine) them in the form $\dfrac{a}{1 + bx^c}$, where a, b and c are arbitrary. We are talking about dollars here so x goes from 0 to less than 100,000. No need to worry about limiting behaviours.
I think one way to do this is to write out the Taylor series expansion on both sides, and then try to match coefficients, but that doesn't look so easy. Another way is pick 3 x's , say x = 100, 4000 and 80000, and try to solve for a, b, c at those points, but it seems there is no solution.
Are there better ways to approach this problem? Thanks!
Curve fitting of the sort you described makes sense for data that does not come from an explicitly given function. Here you already have an elementary function like $\sqrt{x}$ or $\log x$. Fitting $a/(1+bx^c)$ to this will likely to be a bad fit for no gain in simplicity.
To answer your questions directly:
This makes more sense. But a better idea is to take more than 3 values of $x$, somewhat equally distributed (perhaps on logarithmic scale) and fit a curve to them using Least squares.