From half to double, linear to logarithmic scale.

Question

I am making a game where you want a skill value to modify some in game values. With a scale that goes from half to double. 50% to 200%. If I'd do it linear 125% will be the centre but I want the centre to be 100%.

I want a mathematical function:

• Where I input and number from 0 to 100.
• And get out a number from 0.5 to 2 or from 50% to 200%.
• The function maps 0 to 0.5.
• The function maps 50 to 1.
• The function maps 100 to 2.
• And I would like it to follow a logarithmic scale.
• I do not care what happens outside the specified range.

I have made this Google Docs document which shows my plan B.

I hope someone out there can help me this has been a mathematical problem that has bugged me for years.

Answer 1

If $x$ is your input between $0$ and $100$, then $2^{\left(\frac{x}{50}-1\right)}=\exp(\frac{\ln 2}{50}x-1)$ will range from $.5$ to $2$ as specified.