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Proof that there exist only one answer

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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.

 

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