I agree with Half Dome. I have a 7x and 8x, used to have a 10x and have used a zoom going to 15x. I could see in the store that the 15x was impossible to hold still. So, why pay for the extra power? I gave away the 10x because in the field it was also too powerful to hold still. The 7x and 8x work just fine. Now of course, if money is no object you can find stablized binoculars. For about $300 they hold the image very steady.
As to light gathering ability, it is not physics. It is simple math. The amount of light that can be gathered is based on the area of the aperature. The formula is pie R squared, where R is the radius. Double a number then square it and it is four times bigger ( 4 squared = 16, 8 squared = 64, 5 squared = 25, 10 squared = 100).
So, 7x25 binocular has an opening that is 3.14 x 12.5 squared or 490.625 square mm.
A 7x50 binocular has an opening that is 3.14 x 25 squared or 1,962.5 square mm.
(mm = millimeters = 1/1000th of a meter - 25.4mm = 1 inch - 39.4 inches = 1 meter)
So, as you can see, the reason doubling the aperature increases the light gathering ability by a factor of four is because doubling the diameter (or radius which is 1/2 of the diameter) of a circle makes the circle four time bigger. If it is four time bigger, it can gather four times the light.
This also worked for photographs. A 4x5 photograph is 20 square inches. An 8x10 photograph is 80 square inches. A 16x20 photograph is 320 square inches.