Question

Prove that log4(9) is irrational. I have no idea where to even begin on that.

Answer 1

is it log of 9 to the base 4? if it is then here is the solution though i m not sure whether this correct or incorrect....

log3^2 to the base 2^2
so the eqn becomes 2/2log2(3) [log of 3 to the base 2]
=>log2(3)
=>ln3/ln2 [using base change property]
now let this be a rat. no. of the form p/q
=>ln3/ln2=p/q
=>q(ln3)=p(ln2)
=>ln3^q=ln2^p
cancelling ln from both the sides;
3^q=2^p......(1)
But 3^q is always gonna be odd n 2^p is always gonna be even.
So this contradicts the eqn (1).
So log4(9) is irrational.

Answer 2

That's the route I took! Thanks for the reply!