There is something which I don't quite understand, and I'm hoping someone could explain it to me.

Basically, I'm trying to find a statement that expresses this:

$\displaystyle

\log_{ab}x

$

in terms of p and q

$\displaystyle

Let,

$

$\displaystyle

\log_{a}x=p

$

$\displaystyle

\log_{b}x=q

$

$\displaystyle

\log_{a}x= \frac{\log_{ab}x}{\log_{ab}a}=p

$

$\displaystyle

\log_{b}x= \frac{\log_{ab}x}{\log_{ab}b}=q

$

I don't understand where the base ab came from in the above two expressions. Is there a way to do it without the ab?