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

[limitkiller]

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

 

[micromass]

Use this: http://en.wikipedia.org/wiki/Gelfond–Schneider_theorem

 

[limitkiller]

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"

?

 

[micromass]

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

 

[limitkiller]

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

you mean $$\gamma$$ ?

 

[micromass]

you mean $$\gamma$$ ?

Yes.

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

 

[limitkiller]

I get it

 

[limitkiller]

thanks

 

[micromass]

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

 

[limitkiller]

Which is easy, isn't it?

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

 

[micromass]

Which is easy, isn't 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

 

[limitkiller]

I just wanted to point it out in case you missed it

I actually did.