# X^x=4 help me prove.

1. May 19, 2007

### ssb

1. The problem statement, all variables and given/known data
$$x^x=4$$

this is a brain teaser a co-worker gave me. I can take the ln of bothb sides... etc but it ends up going in a circle. any guidance on this problem? I know one solution is 2 but he said there is at least one more solution.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 19, 2007

### cristo

Staff Emeritus
He's incorrect; the only solution to the equation is x=2. I can't think of an elegant proof right now, but you could use the newton-raphson method and observe that no matter your choice for intial solution, the algorithm will always converge to 2.

3. May 19, 2007

### BSMSMSTMSPHD

Using logarithmic differentiation, you can show that the function has only a single critical point (a minimum) at x = 1/e. By observation, x = 2 is a solution. Since there are no other critical points, this must be the only solution.

Last edited: May 19, 2007