# I need a function with the following behavior

Probably a simple question, but my math is a bit rusty... I need a function which looks similar to logarithmic (fast rise at the beginning, then slows down). $f(0) = 0$, $f(b) = a$ ($a = 100$, $b = 2500$ but should be changeable, in between $500$-$5000$)

I don't need a horizontal asymptote because I'll cut off the values when $x$ reaches $b$.

Simply, I need it for my site for calculating a score as a percentage based on number of voters, which rises in a logarithmic fashion until $x$ reaches certain number ($500$-$5000$) and then it bumps the $100$ percent value. (thus giving new users instant gratification, and preventing score spamming for those with more votes).

I played with logarithmic functions but it's hard for me to get something which can be easily parametrized to easily change the parameter $b$ while keeping the flow.

Any ideas?

-
I can understand: f(0)=0 and f(b)=a. What do you mean by "until x reaches certain number (500-5000) and then it bumps the 100 percent value."? How about an example? – NoChance Aug 25 '12 at 11:53
Basically what I meant is to have function be close to 100 on y-axis when x approaches a. It doesn't have to reach 100 precisely in a, because I was cutting it off (accuracy wasn't an issue) – Robert Osswald May 13 '13 at 9:35

Perhaps $a\sqrt{\frac{x}{b}}$ (however it "slows down" much slower than a logarithmic function...)?
How about $\frac{100}{\ln{2501}}\ln{(x+1)}$?
i.e. $\frac{a}{\ln(b+1)}\ln(x+1)$ – You Aug 25 '12 at 12:00