x^x = x Taking logarithm to the base 10,we get x log_10 | x | = log_10 | x | or #160; (x 1 ) log _10 | x | = 0 Hence ...

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Conversion of Numbers Using Logarithms

solve xx =...

Question

solve $x^{x} = x$

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Solution

$x^{x} = x$
Taking logarithm to the base 10,we get
$x l o g_{10} | x | = l o g_{10} | x |$
or $(x - 1) l o g_{10} | x | = 0$
Hence $x = 1$
or $l o g_{10} | x | = 0$ which gives
$| x | = 1$ or $x = \pm 1$
Hence the solution of the given equation is $x = \pm 1$

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