Proof 2^x = x

  • #1
If 2^x = x

then

2^(2^x) = x

also!

2^2^(2^x) = x

2^2^2^(2^x) = x

2^2^2^2^2^2^2^ ...^(2^x)_n = x


As russ wrote to Dr. Math
On 04/03/2004 at 05:43:34 (Eastern Time),
>[Question]
>The equation 2^x = x .
>
>2^x = x
>
>then
>
>2^(2^x) = x
>
>2^2^(2^x) = x
>
>2^2^2^(2^x) = x
>
>2^2^2^2^2^2^2^2^ ... 2^(2^x)_n
>
>How can this equation be solved symbolically?
>
>
>[Difficulty]
>
>
>[Thoughts]
>2^x = x
>
>2 = x^[1/x]
>
>x^[1/x]- 1 - 1 = [2^x]/x ...

Dear Russ,
What is the scope of your investigation? Are you
looking for
answers or methodology? Are you interested in complex
solutions or
only real numbers?
Your thoughts about an infinite series, raising x
to the power x to
the power x are astute, and could be one way to look
for a solution.
There are other methods as well, but none that
involves only a finite
number of calculations. The only solutions to this
kind of equation
are as a limit to an infinite process.
You may be interested in this chapter from our
archives:
http://www.mathforum.org/library/drmath/view/53229.html



 

Answers and Replies

  • #2
NateTG
Science Advisor
Homework Helper
2,450
6
Hmm.
[tex]\frac{d}{dx} 2^x = \ln 2 2^x[/tex]
which is monotone increasing and
[tex]\frac{d}{dx} x = 1[/tex]
which is constant, so if
[tex]2^x > x[/tex]
where
[tex]\ln 2 \times 2^x = 1 \rightarrow x= -\log_2 ({\ln 2}) \approx 0.5[/tex]
then the only solutions are imaginary.
So I don't think there are any real solutions.
 
  • #3
Huh?

If 2^x=x, then

2^(2^x) does not equal x, it equals 2^x; you have to do the same thing to both sides, right??? I guess I don't understand your question then. It definitely does not have real solutions, because of the reason above.
 
  • #4
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,950
19
If 2^x=x, then

2^(2^x) does not equal x, it equals 2^x;

Sure, 2^(2^x) equals 2^x... but what does 2^x equal?
 
  • #5
Ah i see, defined recursively
 
  • #6
1,605
2
x=-LambertW(-log2)/log2, i think...
 

Related Threads on Proof 2^x = x

  • Last Post
Replies
19
Views
20K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
19
Views
2K
Replies
9
Views
13K
  • Last Post
Replies
3
Views
2K
  • Last Post
2
Replies
26
Views
3K
T
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
8
Views
49K
Replies
1
Views
2K
Top