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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=W^(∼) (36)≈2.107036463957
$${x}=\overset{\sim} {\boldsymbol{\mathcal{W}}}\left(\mathrm{36}\right)\approx\mathrm{2}.\mathrm{107036463957} \\ $$
Commented by Tawa11 last updated on 11/Jan/22
Sir, please show workings.
$$\mathrm{Sir},\:\mathrm{please}\:\mathrm{show}\:\mathrm{workings}. \\ $$
Commented by mr W last updated on 11/Jan/22
there is no workings! it can only be approximated. W^∼ (x) is just for fun. mr W has defined this new function as the inverse function of the function f(x)=x^x^x . i.e. W^∼ (x)^(W^∼ (x)^(W^∼ (x)) ) =x. or if y=x^x^x , then x=W^∼ (y). since mr W is a nobody, his new function W^∼ (x) is not yet popular. you are the second person in the word at all who knows about the existence of such a function. keep this for yourself! quite different is it with mr Lambert. he defined a new function W(x) as the inverse function of the function f(x)=xe^x , i.e. W(x)e^(W(x)) =x. but his function W(x) is well known. even you and i know it.
$${there}\:{is}\:{no}\:{workings}!\:{it}\:{can}\:{only}\:{be} \\ $$$${approximated}. \\ $$$$\overset{\sim} {\boldsymbol{\mathcal{W}}}\left({x}\right)\:{is}\:{just}\:{for}\:{fun}.\:{mr}\:\boldsymbol{\mathcal{W}}\:{has} \\ $$$${defined}\:{this}\:{new}\:{function}\:{as}\:{the} \\ $$$${inverse}\:{function}\:{of}\:{the}\:{function} \\ $$$${f}\left({x}\right)={x}^{{x}^{{x}} } .\:{i}.{e}.\:\overset{\sim} {\boldsymbol{\mathcal{W}}}\left({x}\right)^{\overset{\sim} {\boldsymbol{\mathcal{W}}}\left({x}\right)^{\overset{\sim} {\boldsymbol{\mathcal{W}}}\left({x}\right)} } ={x}.\:{or} \\ $$$${if}\:{y}={x}^{{x}^{{x}} } ,\:{then}\:{x}=\overset{\sim} {\boldsymbol{\mathcal{W}}}\left({y}\right).\:{since}\:{mr}\:\boldsymbol{\mathcal{W}} \\ $$$${is}\:{a}\:{nobody},\:{his}\:{new}\:{function}\:\overset{\sim} {\boldsymbol{\mathcal{W}}}\left({x}\right) \\ $$$${is}\:{not}\:{yet}\:{popular}.\:{you}\:{are}\:{the}\:{second} \\ $$$${person}\:{in}\:{the}\:{word}\:{at}\:{all}\:{who}\:{knows}\: \\ $$$${about}\:{the}\:{existence}\:{of}\:{such}\:{a}\: \\ $$$${function}.\:{keep}\:{this}\:{for}\:{yourself}! \\ $$$${quite}\:{different}\:{is}\:{it}\:{with}\:{mr}\:{Lambert}. \\ $$$${he}\:{defined}\:{a}\:{new}\:{function}\:{W}\left({x}\right)\:{as} \\ $$$${the}\:{inverse}\:{function}\:{of}\:{the}\:{function} \\ $$$${f}\left({x}\right)={xe}^{{x}} ,\:{i}.{e}.\:{W}\left({x}\right){e}^{{W}\left({x}\right)} ={x}. \\ $$$${but}\:{his}\:{function}\:{W}\left({x}\right)\:{is}\:{well}\:{known}. \\ $$$${even}\:{you}\:{and}\:{i}\:{know}\:{it}. \\ $$
Commented by Tawa11 last updated on 11/Jan/22
Wow, this is great sir. God will crown your effort sir.
$$\mathrm{Wow},\:\mathrm{this}\:\mathrm{is}\:\mathrm{great}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{will}\:\mathrm{crown}\:\mathrm{your}\:\mathrm{effort}\:\mathrm{sir}. \\ $$
Commented by alephzero last updated on 11/Jan/22
Great sir! But could you clarify the clear definition of the W−function?
$$\mathrm{Great}\:\mathrm{sir}!\:\mathrm{But}\:\mathrm{could}\:\mathrm{you}\:\mathrm{clarify} \\ $$$$\mathrm{the}\:\mathrm{clear}\:\mathrm{definition}\:\mathrm{of}\:\mathrm{the}\:\mathcal{W}−\mathrm{function}? \\ $$
Commented by mr W last updated on 11/Jan/22
do you mean the lambert W function? as i have said, the lambert W function is defined as the inverse function of xe^x , i.e. if y=xe^x , then x=W(y).
$${do}\:{you}\:{mean}\:{the}\:{lambert}\:{W}\:{function}? \\ $$$${as}\:{i}\:{have}\:{said},\:{the}\:{lambert}\:{W}\:{function} \\ $$$${is}\:{defined}\:{as}\:{the}\:{inverse}\:{function} \\ $$$${of}\:{xe}^{{x}} ,\:{i}.{e}.\:{if}\:{y}={xe}^{{x}} ,\:{then}\:{x}={W}\left({y}\right). \\ $$
Commented by alephzero last updated on 12/Jan/22
No! I mean Your function, mr. W! W(x) = ? And what about the name?
$$\mathrm{No}!\:\mathrm{I}\:\mathrm{mean}\:\mathrm{Your}\:\mathrm{function},\:\mathrm{mr}.\:\boldsymbol{\mathcal{W}}! \\ $$$$\mathbb{W}\left({x}\right)\:=\:? \\ $$$$\mathrm{And}\:\mathrm{what}\:\mathrm{about}\:\mathrm{the}\:\mathrm{name}? \\ $$
Commented by mr W last updated on 12/Jan/22
you shouldn′t take it serious! it′s just for fun! but when you want to know, W^∼ (x) is just defined as the inverse function of f(x)=x^x^x , i.e. it fulfills W^∼ (x)^(W^∼ (x)^(W^∼ (x)) ) =x. you can not ask W^∼ (x)=?, because it′s a definition. Name of this useless function? i don′t know. what′s your suggestion?
$${you}\:{shouldn}'{t}\:{take}\:{it}\:{serious}!\:{it}'{s}\:{just} \\ $$$${for}\:{fun}!\: \\ $$$${but}\:{when}\:{you}\:{want}\:{to}\:{know}, \\ $$$$\overset{\sim} {{W}}\left({x}\right)\:{is}\:{just}\:{defined}\:{as}\:{the}\:{inverse} \\ $$$${function}\:{of}\:{f}\left({x}\right)={x}^{{x}^{{x}} } ,\:\:{i}.{e}.\:{it}\:{fulfills} \\ $$$$\overset{\sim} {{W}}\left({x}\right)^{\overset{\sim} {{W}}\left({x}\right)^{\overset{\sim} {{W}}\left({x}\right)} } ={x}.\: \\ $$$${you}\:{can}\:{not}\:{ask}\:\overset{\sim} {{W}}\left({x}\right)=?,\:\:{because}\: \\ $$$${it}'{s}\:{a}\:{definition}. \\ $$$$ \\ $$$${Name}\:{of}\:{this}\:{useless}\:{function}? \\ $$$${i}\:{don}'{t}\:{know}.\:{what}'{s}\:{your}\:{suggestion}? \\ $$
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, W^∼ (x).
$$\mathrm{My}\:\mathrm{suggestion}\:\mathrm{is}\:\mathrm{like}\:“\mathrm{Wetta}−\mathrm{function}'' \\ $$$$\mathrm{And}\:\mathrm{it}'\mathrm{s}\:\mathrm{designation}\:\mathrm{like}\:\mathbb{W}\left({x}\right). \\ $$$$\mathrm{Or},\:\mathrm{if}\:\mathrm{You}\:\mathrm{want},\:\overset{\sim} {{W}}\left({x}\right). \\ $$
Commented by Tawa11 last updated on 15/Jan/22
Sir mrW. Please how did you press or get the approximation.
$$\mathrm{Sir}\:\mathrm{mrW}.\:\:\mathrm{Please}\:\mathrm{how}\:\mathrm{did}\:\mathrm{you}\:\mathrm{press}\:\mathrm{or}\:\mathrm{get}\:\mathrm{the}\:\mathrm{approximation}. \\ $$
Commented by mr W last updated on 15/Jan/22
to approximate graphically you can use many tools like geogebra. as you may have noticed, grapher is almost the only tool i use.
$${to}\:{approximate}\:{graphically}\:{you}\:{can} \\ $$$${use}\:{many}\:{tools}\:{like}\:{geogebra}.\:{as}\:{you} \\ $$$${may}\:{have}\:{noticed},\:{grapher}\:{is}\:{almost} \\ $$$${the}\:{only}\:{tool}\:{i}\:{use}. \\ $$
Commented by Tawa11 last updated on 15/Jan/22
God bless you sir
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$
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...
$$\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} } =\mathrm{36} \\ $$$$\left(\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} } \right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} =\left(\mathrm{6}^{\left(\mathrm{2}\right)\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} \right) \\ $$$$\left(\left(\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \right)^{\boldsymbol{\mathrm{x}}} \right)^{\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}} =\mathrm{6}^{\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}} \\ $$$$\left(\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \right)^{\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}}} =\mathrm{6}^{\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}} \\ $$$$\left(\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \right)=\mathrm{6}^{\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}} \\ $$$$\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \right)=\boldsymbol{\mathrm{ln}}\left(\mathrm{6}^{\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}} \right) \\ $$$$\boldsymbol{\mathrm{xln}}\left(\boldsymbol{\mathrm{x}}\right)=\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{ln}}\left(\mathrm{6}\right) \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{2}} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\mathrm{6}\right) \\ $$$$\boldsymbol{\mathrm{e}}^{\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)} \boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\mathrm{6}\right) \\ $$$$\boldsymbol{\mathrm{e}}^{\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)} \mathrm{2}\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{2}\left(\mathrm{2}\boldsymbol{\mathrm{ln}}\mathrm{6}\right) \\ $$$$\boldsymbol{\mathcal{W}}\left(\boldsymbol{\mathrm{e}}^{\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)} \mathrm{2}\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)\right)=\boldsymbol{\mathcal{W}}\left(\mathrm{2}\left(\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\mathrm{6}\right)\right)\right) \\ $$$$\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)=\boldsymbol{\mathcal{W}}\left(\mathrm{2}\left(\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\mathrm{6}\right)\right)\right) \\ $$$$\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)=\frac{\boldsymbol{\mathcal{W}}\left(\mathrm{2}\left(\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\mathrm{6}\right)\right)\right)}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{x}}=\boldsymbol{\mathrm{e}}^{\left\{\frac{\boldsymbol{\mathcal{W}}\left(\mathrm{2}\left(\mathrm{2}\boldsymbol{\mathrm{ln}}\left(\mathrm{6}\right)\right)\right)}{\mathrm{2}}\right\}} \\ $$$$\boldsymbol{\mathrm{x}}=\mathrm{2}.\mathrm{1582677197900611}… \\ $$
Commented by mr W last updated on 11/Jan/22
x^x^x ≠(x^x )^x =x^(x×x) =x^x^2 therefore that what you got is the root of x^x^2 =36, not the root of x^x^x =36!
$${x}^{{x}^{{x}} } \neq\left({x}^{{x}} \right)^{{x}} ={x}^{{x}×{x}} ={x}^{{x}^{\mathrm{2}} } \\ $$$${therefore}\:{that}\:{what}\:{you}\:{got}\:{is}\:{the} \\ $$$${root}\:{of}\:{x}^{{x}^{\mathrm{2}} } =\mathrm{36},\:{not}\:{the}\:{root}\:{of} \\ $$$${x}^{{x}^{{x}} } =\mathrm{36}! \\ $$
Commented by mr W last updated on 11/Jan/22
example: 4^3^2 =4^9 =262144≠(4^3 )^2 =64^2 =4096
$${example}: \\ $$$$\mathrm{4}^{\mathrm{3}^{\mathrm{2}} } =\mathrm{4}^{\mathrm{9}} =\mathrm{262144}\neq\left(\mathrm{4}^{\mathrm{3}} \right)^{\mathrm{2}} =\mathrm{64}^{\mathrm{2}} =\mathrm{4096} \\ $$

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