What formula to give small values more weight?

Question

I have the following image which contain images ordered and scaled according a rank value, the biggest having a rank of 1 and the smallest a rank of 77:

enter image description here

I would like to give smaller ranks more 'weight', so they become a little bigger.

The logarithm seems quite interesting but the problem is that I don't know how to 'contain' it in a 0.0 to 1.0 space:

enter image description here

Do you know what formula could be used to give smaller values more weight ?

Your purpose being to shift values ranging $1$ thru $77$ to values ranging $0$ thru $1$??? If yes, then how about $\frac{\log(x)}{\log(77)}$? Or $\sqrt{\frac{x}{77}}$? — barak manos, Oct 28 '14 at 15:46
@barakmanos : works great, thank you ! feel free to post an answer so I can mark it. — Eric Cartman, Oct 28 '14 at 16:23

score 3 · Accepted Answer · answered Oct 28 '14 at 16:38

If your purpose is to shift the values from $[1,77]$ to $[0,1]$ while increasing the relative weights of the smaller ones comparing with the larger ones, then you can try either one of the following options:

$\dfrac{\log(x)}{\log(77)}$
$\sqrt{\dfrac{x}{77}}$

What formula to give small values more weight?

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