Q:

Map/Transform linear function to exponent

I have a slider:
min val. = 0
max val. = 20'000
step = 0.1

min, max values and step are constants.
On each step (thumb position change) slider returns a current value and a thumb position in % (see. img).
How can I transform the current returned value to make it grow exponentially like on the 2nd graphic - where 0 - 8'000 - take 80% of slider width.
Red graphic:

On each step (thumb position change) slider returns a current value and a thumb position in % - current value grows linearly depending on the thumb position.

Green graphic: 

This is what I need. I can only use current thumb position and current slider value as a function arguments.

A:

I suggest you try the following function
$$
f(x) = \alpha\tanh(\beta x)
$$
where the constants $\alpha$ and $\beta$ have been selected so that $f(8000) = 0.8$ and $f(20000) = 1$. The values for the constants were determined numerically to be
$$
\alpha = 1.00912774623104, \quad \beta = 0.000134853425544814
$$
Using these values the curve $y = f(x)$ for $x\in [0,20000]$ is given by the following figure.

