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I am trying to simulate a slider on a webpage that behaves non-linearly as shown in the illustration below.

I want to generate a mathematical formula that can represent these states and also give all values in between. I thought this looked like something I can achieve with a log formula, but I could not produce a formula that works.

enter image description here

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You can achieve this with an exponential function.

First define $x$ as a percentage on the slider.

$f(x)=100\times (\sqrt[50]10)^x$

If we evaluate $\sqrt[50]10$, we get:

$f(x)=100\times 1.047129^x$

When $x=50$, $f(50)=1000$. When $x=0$, $f(0)=100$.

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