Carl,

I am not sure exactly what you want but it seems to me that the most important aspect is that the logo not be distorted. In your logo the ratio of the height to the width is 0.75/1.2938 = 0.5797. Hence on any size paper if you specify the width then the height will need to be 0.5797 times the width.

If the width of the scaled logo is w units and the height is h units then its area is w × h square units. But to maintain the ratio of height to width, h = 0.5797 w. Thus the area is h × w = 0.5797 w². Hence if you know the area you want, A, then set A = 0.5797 w² and solve for w.

If I haven't answered your question write back,
Harley

Carl wrote back

Question from Carl:

Harley,
Thank you so much for your response. Unfortunately, I don't think I was clear on what I am trying to do here.
I have the ability to scale the logo without having to do the math... leaving my brain weak and feeble.
My main concern is that the logo always be the correct size on different sized documents. Smaller on smaller documents and larger on poster or billboards.
Thing is, the size of the logo must relate DIRECTLY to the overall size of the paper or document it is placed on.

The standard is...
The logo will always be sized at 1.2938(w) x .75(h) on a letter size page (8.5 x 11).
So... an example problem would be...
What size should the logo be on a trade show graphic that is sized 2 feet by 10 feet.
For a while I have been scaling based upon the longest dimension. Unfortunately, this will often times create a logo that is HUGE for the document.
I hope this makes more sense!
Thank again,
Carl

Carl,

I am taking your statement "the size of the logo must relate DIRECTLY to the overall size of the paper or document it is placed on" to mean that the logo must occupy the same proportion of the area of the document as it does on an 8.5 by 11 page.

Suppose the logo has dimensions w by h on a page of area a square units and you want to scale it to dimensions W by H on a page of A square units. (In your case w = 1.2938 inches, h = 0.75 inches and a = 8.5 × 11 = 93.5 square inches.) I still claim that you want the ratio of w to h to be the same as the ratio of W to H so there is a constant k so that h = kw and H = kW. Thus the area of the unscaled logo is kw² and the area of the scaled logo is kW². Hence the unscaled logo occupies kw²/a of the page and the scaled logo occupies kW²/A of its page. If these are the same then

kw²/a = kW²/A

and thus

kW²/kw² = W²/w² = A/a

or

W = √(A/a) w

Thus the scale factor is √(A/a) which in your situation is √(A/93.5).

Watch out for the units. I think it is easiest to put all the lengths in the same units. Thus in your example of sizing up to 2 feet by 10 feet I would use 24 inches by 120 inches. Thus A = 24 × 120 = 2880 square inches and hence the scale factor is

√(A/a) = √(2880/93.5) = √30.802139 = 5.550

This will give W in inches.

You still may have to fine tune this using your experience as a graphic designer. For a ridiculous example suppose the trade show banner is 4 inch by 720 inches. Its area is still 2880 square inches and the scaled height is 5.550 × 0.75 = 4.16 inches which won't even fit on the banner.

I hope this helps,
Harley