X^X^X^... = 2

Date: 05/17/2000 at 20:13:49
From: Dominic
Subject: limits

Solve for x:

     x^x^x^... = 2

I tried to use limits, tried substituting various values, and tried 
graphing the function on the graphing calculator, but got no solution. 
Help!

Date: 05/17/2000 at 23:07:30
From: Doctor Douglas
Subject: Re: limits

Hi Dominic,

Thanks for submitting your question to Ask Dr. Math.

This is a very interesting problem. Before answering it, let's make 
sure that we mean the same thing by x^x^x^...  My convention is that 
we do all powers in the exponent before raising a base to that 
exponent, meaning, for example, that:

     4^3^2 = 4^(3^2) = 4^9 = 262144

Now let's see if we can make progress on this problem. Let's let:

     y = x^x^x^...    provided that this limit exists (assume it does)

Then:

     y = 2            of course we know y; what's x?
 
But we also see that

     x^(y) = 2        seems like magic, the tower collapses instantly!
      x^2  = 2
       x   = sqrt(2)

Check this on a calculator, and see if it converges. (According to my 
calculator, it does.)

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/