How to prove that $\frac{\mathrm{ln}12}{\mathrm{ln}18}$ is irrational witout using the change of base rule?

I have to show that $\frac{\mathrm{ln}12}{\mathrm{ln}18}$ is irrational by using change of base rule.

At the beginning I have proved that $\mathrm{ln}r$ is irrational for any rational $r$, $r\ne 1$. Then using this we can say that $\mathrm{ln}12$ and $\mathrm{ln}18$ are irrational.

But from here it is difficult for me to show that the fraction is irrational knowing that both the numerator and the denominator are irrational.

I have to show that $\frac{\mathrm{ln}12}{\mathrm{ln}18}$ is irrational by using change of base rule.

At the beginning I have proved that $\mathrm{ln}r$ is irrational for any rational $r$, $r\ne 1$. Then using this we can say that $\mathrm{ln}12$ and $\mathrm{ln}18$ are irrational.

But from here it is difficult for me to show that the fraction is irrational knowing that both the numerator and the denominator are irrational.