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What is the value of $x$ if $$x^x = x?$$ Can somebody show step by step please. Thanks!

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    What about $1$? – 2017-02-07
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    x is either 1 or -1. – 2017-02-07

2 Answers 2

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Given:$\;\;$$x^{x} = x\;,\;$

Taking the logarithms on both the sides of equation we get

$ x\times\log (|x|) = \log (|x|)$

$ \therefore \;\: (x - 1)\times\log (|x|) = 0$

For the above equation to be true

Either $\;\;$$x-1 = 0\;\;$ or $ \;\;$$\log (|x|) =0$

Therefore $\;\;$$x = 1\;\;$ or $\;\;$$|x| = 1$.

Hence, the solution is $\;\;$$x = 1\;\;$ or $\;\;$$x = -1$.

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Clearly, $x\ne0$

So, we have $$x^{x-1}=1$$

For $a^b=1$

either $a=1$

or $a=-1,b$ even

or $b=0$

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    @Alex, Exactly. That was my first statement, right? – 2017-02-07
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    How does one know those are the only solutions to $a^b =1$? – 2017-02-07
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    @fleablood, If $$a>1, b>0, a^b>1$$ and for $$a>1, b<0, a^b<1$$ and so on – 2017-02-07
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    And how do you know *that*? – 2017-02-07