Consider the following simple problem:Given fractions $a/b$, $c/d$ with $a/b<1$ and $c/d<1$, and natural  number $e$, both encoded in binary (so the problem size is  $\log(a)+\log(b)+\log(c)+\log(d)+\log(e)$, decide whether  $$(\frac{a}{b})^e \geq \frac{c}{d}$$.What's the precise complexity of this problem? Note that one can simply compute $(\frac{a}{b})^e$, as it might require $e\cdot (\log(a)+\log(b))$ bits to store. Any upper/lower bound? Do I miss something here?
The complexity of comparing a power to some threshold
cc.complexity theory