Proof that there exist only one answer

[BMAD 63]

Prove that x^(1/x) = x^-x  has only one solution

[bonanova 81]

Maybe, maybe not.

Spoiler

Taking the natural log of both sides yields

(1/x) ln(x) = (-x) ln(x)

ln(x) [1/x + x] = 0

This has solutions

ln(x) = 0 ==> x = 1

And

1/x + x = 0 ==> x² + 1 = 0 ==> x = (+/-) i

So there are three solutions, one of which is real.