Find if the rational log of a rational number is rational?

Question

I'm currently working on a project and need to find whether or not $\log_\frac{a}{b}\Bigl(\frac{c}{d}\Bigr)$ can be expressed as $\frac{f}{g}$. Is there a computationally efficient way to do this? Also if it can is there a way to calculate $f$ and $g$?

Answer 1

Assume $\log_{a/b}(c/d) = f/g$. Raisin $a/b$ to both sides gives

$c/d = (a/b)^{(f/g)}$

Which can be rewritten as

$c^gb^f = a^fd^g$

Assuming the fractions where written in reduced form so that $c,d$ are coprime and $a,b$ are coprime, we get that $c^g = a^f$ and $b^f = d^g$.

From here it's not difficult to find $f$ and $g$.