A scaling factor for a logo that decreases as page gets bigger

Question

I'm Okay with fixed scaling factors but this has got me beat.

I have a page that is 100mm and a logo that is 25mm wide making it 25% of the page width when the logo is scaled at 100%.

As the page size increases I want to enlarge the Logo also - but not proportionately - so that when the page is incrementally enlarged to a maximum of 500% or 500mm the logo is scaled to a maximum of say 75mm (300%) not 125mm (500%). Basically as the page size increases the logo enlargement increase to a lesser extent.

Any help would be gratefully received

Thanks

Answer 1

Suppose the page width is $P$ millimetres and the logo width is $L$ millimetres

One of many possible solutions to your objective could be an curve $$L=25\left(\frac{P}{100}\right)^{\frac{\log(75)-\log(25)}{\log(500)-\log(100)}} \approx 1.07827\times P^{0.6826}$$ which grows with $P$ but sublinearly

It would suggest the following logo widths for various page widths:

Page (mm)   Logo (mm)     
  10          5.2
  20          8.3
  50         15.6
 100         25.0
 200         40.1
 300         52.9
 400         64.4
 500         75.0
1000        120.4