Working With Large Exponents

Since logarithms are monotonically increasing, you can determine if $$a^b > c^d$$ by instead checking if $$\log(a^b) > \log(c^d)$$ which can be rewritten (using basic properties of logarithms) as $$b\log(a) > d\log(c)$$ This results in the comparison of two numbers on the order of those given in the input (that is, both numbers will be within an order of magnitude or so of $a,b,c,d$), which is definitely feasible on a computer.

You can also use logs to calculate other properties, as long as you remember to convert your solution back to the original numbers (by means of an exponential) when you're done. For example, to compute whether $${a_1}^{b_1}{a_2}^{b_2} > {c_1}^{d_1}{c_2}^{d_2}$$, you can take the log of both sides to get \begin{align} \log({a_1}^{b_1}{a_2}^{b_2}) &> \log({c_1}^{d_1}{c_2}^{d_2}) \\ \log({a_1}^{b_1}) + \log({a_2}^{b_2}) &> \log({c_1}^{d_1}) + log({c_2}^{d_2}) \\ b_1\log(a_1) + b_2\log(a_2) &> d_1\log(c_1) + d_2\log(c_2) \\ \end{align}

which is again computationally feasible.