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Question Number 204518 by Abdullahrussell last updated on 20/Feb/24

	Answered by TonyCWX08 last updated on 20/Feb/24

	$l e t u s s o l v e y^{y} = 2 f i r s t$ $y^{y} = 2$ $y = 2^{\frac{1}{y}}$ $y = {(e^{l n (2)})}^{\frac{1}{y}}$ $y = e^{\frac{l n (2)}{y}}$ $l n (2) = \frac{l n (2)}{y} e^{\frac{l n (2)}{y}}$ $U s i n g L a m b e r t W f u n c t i o n,$ $W (l n (2)) = \frac{l n (2)}{y}$ $y = \frac{l n (2)}{W (l n (2))}$ $y = e^{W (l n (2))}$ $A c c o r d i n g t o W o l f r a m A l p h a,$ $e^{W (l n (2))} \approx 1.55961 = y$ $S o$ $y^{y} = 2$ $1 . 55961^{1.55961} = 2$ $t a n {(x)}^{t a n (x)} = 1 . 55961^{1.55961}$ $t a n (x) = 1.55961$ $x = \tan^{- 1} (1.55961)$ $x = 1.00064 + k π, k \in Z$
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