You may start with something like

fc(x) = (log(x+c) - log(c)) / (x + c)

This is a positive function (for x > 0) that aproaches 0 at infinity, is 0 at 0, and has a single maximum. This is true for any c > 0.

Now you can consider, say, n·fc(m·x) with three parameters to choose the most suitable function. For b, you must be able to choose one of the three or some combination thereof by relating their values.

Actually, the base of the logarithm is a fourth parameter.