Infinite exponential

  • #1
Consider the equation

[tex]x^{x^{x^{...}}} = 2 [/tex]

Does x exist ?

Well, at first, I would say x=sqrt(2) , but is this ok?

In general, x^x^x^... = Z would imply x = Z^(1/Z)
But, if x>1, then x^x^x^... is crescent.
In other words, when "Z" increases, then "x" increases.

However, lim oo Z^(1/Z) = 1
So, if x^x^x^... increases, it means that "x" goes to 1 ?!

Where is the mistake?
 

Answers and Replies

  • #2
270
0
x must be between [tex]e^{-e}[/tex] and [tex]e^{\frac{1}{e}}[/tex]. Euler gave a proof showing that it only has a limit in this interval.

I don't know if the answer is right, but it's in the interval... so it's possible.
 
Last edited:
  • #3
39
0
I reduced it to
x^k * ln(x) = ln(2) for some k
so i guess find x and k? like x= 2 and k = 0....
dunno if this right just wondering.....
 
  • #4
morphism
Science Advisor
Homework Helper
2,015
4
It can be reduced to x^2 = 2 immediately.
 
  • #5
39
0
It can be reduced to x^2 = 2 immediately.
could u explain for some reason I am not able to see it?
Thanks!
 
  • #6
1,030
4
[tex]x^{x^{x...}}}=x^2=2[/tex]
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
41,833
961
In case you are wondering, Dragonfall is taking x to the power of each side.
x to the [itex]x^{x^{x...}}}[/itex] is equal to [itex]x^{x^{x...}}}[/itex] since there were an infinite number of "x"s to begin with and according to the equation, that is equal to 2. On the right side, of course, x to the 2 power is x2: 2= x2.
 
  • #8
39
0
In case you are wondering, Dragonfall is taking x to the power of each side.
x to the [itex]x^{x^{x...}}}[/itex] is equal to [itex]x^{x^{x...}}}[/itex] since there were an infinite number of "x"s to begin with and according to the equation, that is equal to 2. On the right side, of course, x to the 2 power is x2: 2= x2.
makes sense overlooked the fact that there are infinite number of them..
thanks.
 

Related Threads on Infinite exponential

Replies
6
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
5K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
21
Views
7K
  • Last Post
Replies
4
Views
4K
  • Last Post
Replies
4
Views
8K
  • Last Post
Replies
1
Views
9K
Top