## 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"
?

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

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

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

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 Quote by limitkiller 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
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus 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?

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