Proving x^x=4: A Brain Teaser with Multiple Solutions | Homework Help

[ssb]

Homework Statement

$$x^x=4$$

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.

 

[cristo]

I know one solution is 2 but he said there is at least one more solution.

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.

 

[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.