# Thread: Prove that z=log base p (qr) is an irrational number

1. ## Prove that z=log base p (qr) is an irrational number

Hello everyone,
I'd like to prove that, providing p q and r are distinct but arbitrary prime numbers that z=logp(qr) is an irrational number. I know it's supposed to be a proof by contradiction but I'm having tons of problems getting through it. Could anyone help?

2. Originally Posted by JTL4869
Hello everyone,
I'd like to prove that, providing p q and r are distinct but arbitrary prime numbers that z=logp(qr) is an irrational number. I know it's supposed to be a proof by contradiction but I'm having tons of problems getting through it. Could anyone help?
Suppose otherwise, then there exist (coprime) integers $a$ and $b$ such that:

$\dfrac{a}{b}=\log_p(qr)$

or:

$(p)^a=(qr)^b$

and you should be able to get your contradiction from the fundamental theorem of arithmetic

CB

3. Thank you! I'm assuming the contradiction is from the uniqueness part of the FTA?

4. Originally Posted by JTL4869
Thank you! I'm assuming the contradiction is from the uniqueness part of the FTA?
Yes.

CB