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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