If you keep everything else the same, it is exponential based on length of print or linear based on area of print. So doubling the dimensions from 10x8 to, for example 20x16 gives you four times the area which the same light is covering so it is getting a quarter of the strength. Therefore multiply the exposure time by four.
10x8 and and 12x16 are not the same aspect ratio so you could do a bit of maths to work it out. If we assume that the same linear length of image which fits the 10" side of the 10x8 is also going to fit on the 16" side of the 12x16 then we can do this:
Exposure multiplier = (large size squared)/(small size squared) = (16x16)/(10x10) = 256/100 = 2.56
So about two and a half times the original exposure will get you in the right sort of area.