# Homework Help: Larson 3.3.25

1. Jan 4, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
Determine all positive rational solutions of x^y=y^x.

2. Relevant equations

3. The attempt at a solution
Obviously, x=y will always work. I think that is the only solution. If I can show that x^y must be rational, I think it will be easy because then both x and y must have the same primes in the numerator and the denominator. I tried writing out x=r/s, y = t/u, and manipulating, which leads to
$$r^{ts} u^{ru} = s^{ts} t^{ru}$$
which is really not helpful.

2. Jan 4, 2008

### dynamicsolo

How about x = 2 , y = 4 ?

3. Jan 4, 2008

### Shooting Star

> I think that is the only solution.

What about 2^4 = 4^2...

4. Jan 5, 2008

### ehrenfest

Hmmm. Maybe that is the only exception. But can we prove that we have found them all....

5. Jan 5, 2008

### morphism

This has been discussed in these forums a few times before, but I can't seem to find the threads. I did find this however.