How can I manipulate a logarithmic function to both grow faster as well as slow down faster?
I need to map a set of values to a range of real numbers between $0$ and $1$ (representing percentages by which to manipulate something). I am using this formula for it ($max > 0$, otherwise the formula would be more complicated):
$$f(x) = \frac {\log (1+x)}{\log (1+max)}$$
lowering the base of the log-function in the dividend makes $f(x)$ grow faster, but then $f(max) = 1$ is not true anymore.
What I am trying to accomplish is that smaller values in my set get mapped to higher percentage values, while higher values get only a slightly higher percentage-increase
You could add increase the additive constant $1$ in both logarithms or, equivalently, multiply both $x$ and $\max$ by some factor.