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

  • #1
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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Answers and Replies

  • #3


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"
?
 
  • #4
22,089
3,291


What is
[tex]\alpha^{\log(\gamma)/\log(\alpha)}[/tex]
?
 
  • #5


What is
[tex]\alpha^{\log(\gamma)/\log(\alpha)}[/tex]
?
you mean [tex]\gamma [/tex] ?
 
  • #6
22,089
3,291


you mean [tex]\gamma [/tex] ?
Yes.

So, IF [itex]\log(\gamma)/\log(\alpha)[/itex] were an algebraic nonrational number, then by applying Gelfond-Schneider we get ...
 
  • #9
22,089
3,291


Of course, to be able to apply Gelfond-Schneider to [itex]\log(2)/\log(3)[/itex], we must first prove that it's not rational...
 
  • #10


Which is easy, isnt it?

since 2 ^a for integer a is never a power of 3...
Right?
 
  • #11
22,089
3,291


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 :smile:
 
  • #12


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

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