Is the ratio of two natural logarithms irrational or rational?

Question

Is there any way to prove that the ratio of two natural logarithms is rational or irrational? Take the natural logarithms of $a = 25$ and $b = 6$, for example. Can you prove $\ln(a)/\ln(b)$ rational or irrational?

Answer 1

$\ln(a)/\ln(b) = n/d$ (where $n,d$ are integers) iff $a^d = b^n$. If $a$ and $b$ are positive integers, this can be decided using the prime factorizations of $a$ and $b$: $a$ and $b$ have the same prime factors, and the ratios of the exponents are the same. In your example, $25$ has prime factor $5$ but $6$ does not, so the answer is no.

Answer 2

$ln(a)/ln(b)=p/q$ implies $qln(a) =pln(b)$ so $e^{pln(b)}=e^{qln(a)}$ which means $b^p=a^q$