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Question Number 58877 by Pk1167156@gmail.com last updated on 01/May/19

Commented by Pk1167156@gmail.com last updated on 01/May/19
solve for x!!
Commented by MJS last updated on 01/May/19

$a^{b^{c}} = a^{(b^{c})} \neq {(a^{b})}^{c} = a^{b c}$ $for {(x^{\sqrt{x}})}^{\sqrt{2}} = x^{\sqrt{2 x}} = \frac{1}{\sqrt{2}} we get x \approx . 0013868 \lor x \approx . 75412$ $but for x^{{(\sqrt{x})}^{\sqrt{2}}} = x^{(x^{\frac{\sqrt{2}}{2}})} = \frac{1}{\sqrt{2}} we get x \approx . 045163 \lor x \approx . 61255$
	Answered by mr W last updated on 01/May/19
	$let t=x^(√x) t^(√2) =(1/(√2)) (√2) ln t=−(1/2)ln 2 ln t=−((ln 2)/(2(√2))) t=e^(−((ln 2)/(2(√2)))) x^(√x) =t (√x) ln x=ln t=−((ln 2)/(2(√2))) let s=(√x) 2s ln s=−((ln 2)/(2(√2))) ln s^s =−((ln 2)/(4(√2))) s^s =e^(−((ln 2)/(4(√2)))) =a s=a^(1/s) =e^((ln a)/s) (1/s)e^((ln a)/s) =1 ((ln a)/s)e^((ln a)/s) =ln a ((ln a)/s)=W(ln a)=Lambert function s=((ln a)/(W(ln a)))=((−((ln 2)/(4(√2))))/(W(−((ln 2)/(4(√2)))))) x=s^2 ⇒x=[−(((ln 2)/(4(√2)))/(W(−((ln 2)/(4(√2))))))]^2 ≈ { (((((−((ln 2)/(4(√2))))/(−0.14110091)))^2 =0.754121)),(((((−((ln 2)/(4(√2))))/(−3.29038565))))^2 =0.001387)) :}$
	$l e t t = x^{\sqrt{x}}$ $t^{\sqrt{2}} = \frac{1}{\sqrt{2}}$ $\sqrt{2} \ln t = - \frac{1}{2} \ln 2$ $\ln t = - \frac{\ln 2}{2 \sqrt{2}}$ $t = e^{- \frac{\ln 2}{2 \sqrt{2}}}$ $x^{\sqrt{x}} = t$ $\sqrt{x} \ln x = \ln t = - \frac{\ln 2}{2 \sqrt{2}}$ $l e t s = \sqrt{x}$ $2 s \ln s = - \frac{\ln 2}{2 \sqrt{2}}$ $\ln s^{s} = - \frac{\ln 2}{4 \sqrt{2}}$ $s^{s} = e^{- \frac{\ln 2}{4 \sqrt{2}}} = a$ $s = a^{\frac{1}{s}} = e^{\frac{\ln a}{s}}$ $\frac{1}{s} e^{\frac{\ln a}{s}} = 1$ $\frac{\ln a}{s} e^{\frac{\ln a}{s}} = \ln a$ $\frac{\ln a}{s} = W (\ln a) = L a m b e r t f u n c t i o n$ $s = \frac{\ln a}{W (\ln a)} = \frac{- \frac{\ln 2}{4 \sqrt{2}}}{W (- \frac{\ln 2}{4 \sqrt{2}})}$ $x = s^{2}$ $\Rightarrow x = {[- \frac{\frac{\ln 2}{4 \sqrt{2}}}{W (- \frac{\ln 2}{4 \sqrt{2}})}]}^{2} \approx {\begin{cases} {(\frac{- \frac{\ln 2}{4 \sqrt{2}}}{- 0.14110091})}^{2} = 0.754121 \\ {(\frac{- \frac{\ln 2}{4 \sqrt{2}}}{- 3.29038565)})}^{2} = 0.001387 \end{cases}$
	Commented by otchereabdullai@gmail.com last updated on 01/May/19

	$Brilliant prof W i would like to name$ $my incoming Baby after you so please if$ $interested send me your name through$ $my e - mail thus my user name thanks$
	Commented by mr W last updated on 01/May/19

	$T h a n k y o u f o r y o u r k i n d n e s s s i r!$ $S o r r y, b u t I {d o n}^{'} t h a v e s u c h a n a m e$ $w h i c h i s n i c e e n o u g h f o r y o u r b a b y .$ $N e v e r t h e l e s s I s e n d y o u r i n c o m i n g$ $b a b y a l l m y b e s t w i s h e s!$
	Commented by Tawa1 last updated on 01/May/19

	$Nice solution sir . God bless you always$
	Commented by Tawa1 last updated on 01/May/19

	$Sir, see question 58899$
	Commented by Pk1167156@gmail.com last updated on 01/May/19
	very nice sir!! God bless you!!
	Commented by otchereabdullai@gmail.com last updated on 01/May/19

	$Ok thanks prof W$
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