Prove $pq + 5(p - q) = 1$

Question

Could you please explain me how to solve:

If $p\:=\:\log _{12}\left(18\right)$ and $q\:=\:\log _{24}\left(54\right)$, $pq\:+\:5\left(p-q\right)\:=\:1$

I tried this way:

$p = \frac{2\log\left(3\right)+\log\left(2\right)}{2\log\left(2\right)+\log\left(3\right)}$, $q = \frac{3\log\left(3\right)+\log\left(2\right)}{3\log\left(2\right)+\log\left(3\right)}$

But not sure what to do next? I'd be grateful if you can help me!

Answer 1

HINT:

The idea is to eliminate $\log2, \log3$

$$(2p-1)\log2=(2-p)\log3$$

$$(3q-1)\log2=(3-q)\log3$$

Divide the the first relation by the second and rearrange.

Answer 2

$ 12 ^ p = 18$ Equation 1

$ 24 ^ q = 54 $ Equation 2

(1) *3 = (2) ,You can write 12,24 and 18 as product of powers of 2 and 3 and then equate exponents on both side