The answer is $5^{4321} < 4^{5321}$. This can be shown by using the ratio method:

$$\frac{5^{4321}}{4^{5321}}=\left(\frac{5^4}{4^5}\right)^{1000}\times\left(\frac{5}{4}\right)^{321}=\left(\frac{625}{1024}\right)^{1000}\times\left(\frac{5}{4}\right)^{321}=\left(\frac{625}{1024}\right)^{679}\times\left(\frac{625}{1024}\right)^{321}\times\left(\frac{5}{4}\right)^{321}<\left(\frac{625\times 5}{1024\times 4}\right)^{321}=\left(\frac{3125}{4096}\right)^{321} < 1$$