- #1

- 80

- 0

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

Last edited:

- Thread starter limitkiller
- Start date

- #1

- 80

- 0

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

Last edited:

- #2

- 22,089

- 3,291

- #3

- 80

- 0

thanks, but how is

equivalent to"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."

?"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

- 80

- 0

you mean [tex]\gamma [/tex] ?What is

[tex]\alpha^{\log(\gamma)/\log(\alpha)}[/tex]

?

- #6

- 22,089

- 3,291

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

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

- #7

- 80

- 0

I get it

- #8

- 80

- 0

thanks

- #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

- 80

- 0

Which is easy, isnt it?

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

Right?

- #11

- 22,089

- 3,291

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 itWhich is easy, isnt it?

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

Right?

- #12

- 80

- 0

- Replies
- 6

- Views
- 4K

- Replies
- 10

- Views
- 3K

- Replies
- 6

- Views
- 5K

- Replies
- 16

- Views
- 3K

- Replies
- 2

- Views
- 2K

- Last Post

- Replies
- 8

- Views
- 2K

- Replies
- 12

- Views
- 2K

- Last Post

- Replies
- 8

- Views
- 2K

- Replies
- 5

- Views
- 8K

- Replies
- 3

- Views
- 2K