Associated Topics || Dr. Math Home || Search Dr. Math

### 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,

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/
```
Associated Topics:
High School Functions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search