$2017^{2027} ~?~ 2027^{2017}$

Let x = 2017^2027, y = 2027^2017. Then log(x) = 2027 log(2017) = 6698.6...., log(y) = 2017 log(2027) = 6669.9..... Ed Gray

One way to do this would be to take the natural logarithm of both to obtain $2027\ln(2017)$ and $2017\ln(2027)$ , and use a Taylor series to approximate the natural logarithm out to enough terms until you could clearly tell which was larger.