# How do we know if Log(2)_3 is not equal to something like ((x^y)+a)

How do we know if Log(2)_3 is not equal to something like ((x^y)+a) ,for rational a,x,y ?

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thanks, but how is
"If α and β are algebraic numbers with α ≠ 0,1 and if β is not a rational number, then any value of αβ = exp(β log α) is a transcendental number."
equivalent to
"if α and γ are nonzero algebraic numbers, and we take any non-zero logarithm of α, then (log γ)/(log α) is either rational or transcendental"
?

What is
$$\alpha^{\log(\gamma)/\log(\alpha)}$$
?

What is
$$\alpha^{\log(\gamma)/\log(\alpha)}$$
?
you mean $$\gamma$$ ?

you mean $$\gamma$$ ?
Yes.

So, IF $\log(\gamma)/\log(\alpha)$ were an algebraic nonrational number, then by applying Gelfond-Schneider we get ...

I get it

thanks

Of course, to be able to apply Gelfond-Schneider to $\log(2)/\log(3)$, we must first prove that it's not rational...

Which is easy, isnt it?

since 2 ^a for integer a is never a power of 3...
Right?

Which is easy, isnt it?

since 2 ^a for integer a is never a power of 3...
Right?
Yeah, that's it (except for a=0 of course, but that's also not possible). I just wanted to point it out in case you missed it

I just wanted to point it out in case you missed it
I actually did.