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Question Number 163833 by mathlove last updated on 11/Jan/22

Commented by mr W last updated on 11/Jan/22

$x = \tilde{W} (36) \approx 2.107036463957$
Commented by Tawa11 last updated on 11/Jan/22

$Sir, please show workings .$
Commented by mr W last updated on 11/Jan/22

$t h e r e i s n o w o r k i n g s! i t c a n o n l y b e$ $a p p r o x i m a t e d .$ $\tilde{W} (x) i s j u s t f o r f u n . m r W h a s$ $d e f i n e d t h i s n e w f u n c t i o n a s t h e$ $i n v e r s e f u n c t i o n o f t h e f u n c t i o n$ $f (x) = x^{x^{x}} . i . e . \tilde{W} {(x)}^{\tilde{W} {(x)}^{\tilde{W} (x)}} = x . o r$ $i f y = x^{x^{x}}, t h e n x = \tilde{W} (y) . s i n c e m r W$ $i s a n o b o d y, h i s n e w f u n c t i o n \tilde{W} (x)$ $i s n o t y e t p o p u l a r . y o u a r e t h e s e c o n d$ $p e r s o n i n t h e w o r d a t a l l w h o k n o w s$ $a b o u t t h e e x i s t e n c e o f s u c h a$ $f u n c t i o n . k e e p t h i s f o r y o u r s e l f!$ $q u i t e d i f f e r e n t i s i t w i t h m r L a m b e r t .$ $h e d e f i n e d a n e w f u n c t i o n W (x) a s$ $t h e i n v e r s e f u n c t i o n o f t h e f u n c t i o n$ $f (x) = {x e}^{x}, i . e . W (x) e^{W (x)} = x .$ $b u t h i s f u n c t i o n W (x) i s w e l l k n o w n .$ $e v e n y o u a n d i k n o w i t .$
Commented by Tawa11 last updated on 11/Jan/22

$Wow, this is great sir . God will crown your effort sir .$
Commented by alephzero last updated on 11/Jan/22

$Great sir! But could you clarify$ $the clear definition of the W - function ?$
Commented by mr W last updated on 11/Jan/22

$d o y o u m e a n t h e l a m b e r t W f u n c t i o n ?$ $a s i h a v e s a i d, t h e l a m b e r t W f u n c t i o n$ $i s d e f i n e d a s t h e i n v e r s e f u n c t i o n$ $o f {x e}^{x}, i . e . i f y = {x e}^{x}, t h e n x = W (y) .$
Commented by alephzero last updated on 12/Jan/22

$No! I mean Your function, mr . W!$ $W (x) = ?$ $And what about the name ?$
Commented by mr W last updated on 12/Jan/22

$y o u {s h o u l d n}^{'} t t a k e i t s e r i o u s! {i t}^{'} s j u s t$ $f o r f u n!$ $b u t w h e n y o u w a n t t o k n o w,$ $\tilde{W} (x) i s j u s t d e f i n e d a s t h e i n v e r s e$ $f u n c t i o n o f f (x) = x^{x^{x}}, i . e . i t f u l f i l l s$ $\tilde{W} {(x)}^{\tilde{W} {(x)}^{\tilde{W} (x)}} = x .$ $y o u c a n n o t a s k \tilde{W} (x) = ?, b e c a u s e$ ${i t}^{'} s a d e f i n i t i o n .$ $N a m e o f t h i s u s e l e s s f u n c t i o n ?$ $i {d o n}^{'} t k n o w . {w h a t}^{'} s y o u r s u g g e s t i o n ?$
Commented by alephzero last updated on 12/Jan/22

$My suggestion is like ‘ ‘ Wetta - {function}^{″}$ $And {it}^{'} s designation like W (x) .$ $Or, if You want, \tilde{W} (x) .$
Commented by Tawa11 last updated on 15/Jan/22

$Sir mrW . Please how did you press or get the approximation .$
Commented by mr W last updated on 15/Jan/22

$t o a p p r o x i m a t e g r a p h i c a l l y y o u c a n$ $u s e m a n y t o o l s l i k e g e o g e b r a . a s y o u$ $m a y h a v e n o t i c e d, g r a p h e r i s a l m o s t$ $t h e o n l y t o o l i u s e .$
Commented by Tawa11 last updated on 15/Jan/22

$God bless you sir$
	Answered by Gbenga last updated on 11/Jan/22
	$x^x^x =36 (x^x^x )^(1/x) =(6^((2)(1/x)) ) ((x^x )^x )^(1/x) =6^(2/x) (x^x )^(x/x) =6^(2/x) (x^x )=6^(2/x) ln(x^x )=ln(6^(2/x) ) xln(x)=(2/x)ln(6) x^2 ln(x)=2ln(6) e^(2ln(x)) ln(x)=2ln(6) e^(2ln(x)) 2ln(x)=2(2ln6) W(e^(2ln(x)) 2ln(x))=W(2(2ln(6))) 2ln(x)=W(2(2ln(6))) ln(x)=((W(2(2ln(6))))/2) x=e^({((W(2(2ln(6))))/2)}) x=2.1582677197900611...$
	$x^{x^{x}} = 36$ ${(x^{x^{x}})}^{\frac{1}{x}} = (6^{(2) \frac{1}{x}})$ ${({(x^{x})}^{x})}^{\frac{1}{x}} = 6^{\frac{2}{x}}$ ${(x^{x})}^{\frac{x}{x}} = 6^{\frac{2}{x}}$ $(x^{x}) = 6^{\frac{2}{x}}$ $\ln (x^{x}) = \ln (6^{\frac{2}{x}})$ $xln (x) = \frac{2}{x} \ln (6)$ $x^{2} \ln (x) = 2 \ln (6)$ $e^{2 \ln (x)} \ln (x) = 2 \ln (6)$ $e^{2 \ln (x)} 2 \ln (x) = 2 (2 \ln 6)$ $W (e^{2 \ln (x)} 2 \ln (x)) = W (2 (2 \ln (6)))$ $2 \ln (x) = W (2 (2 \ln (6)))$ $\ln (x) = \frac{W (2 (2 \ln (6)))}{2}$ $x = e^{{\frac{W (2 (2 \ln (6)))}{2}}}$ $x = 2.1582677197900611 . . .$
	Commented by mr W last updated on 11/Jan/22

	$x^{x^{x}} \neq {(x^{x})}^{x} = x^{x \times x} = x^{x^{2}}$ $t h e r e f o r e t h a t w h a t y o u g o t i s t h e$ $r o o t o f x^{x^{2}} = 36, n o t t h e r o o t o f$ $x^{x^{x}} = 36!$
	Commented by mr W last updated on 11/Jan/22

	$e x a m p l e :$ $4^{3^{2}} = 4^{9} = 262144 \neq {(4^{3})}^{2} = 64^{2} = 4096$
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