Question Number 58899 by Tawa1 last updated on 01/May/19
$$\boldsymbol{\mathrm{S}}\mathrm{olve}\:\mathrm{for}\:\:\mathrm{x}:\:\:\:\:\:\mathrm{x}^{\mathrm{x}^{\mathrm{x}} \:\:} =\:\:\mathrm{729} \\ $$
Commented by mr W last updated on 01/May/19
$${you}\:{can}'{t}\:{solve}\:{this}\:{using}\:{Lambert}\:{function}. \\ $$$${but}\:{you}\:{can}\:{solve}\:{x}^{{x}^{{x}^{{x}} } } =\mathrm{729}. \\ $$
Commented by Tawa1 last updated on 01/May/19
$$\mathrm{Ohh},\:\mathrm{why}\:\mathrm{it}\:\mathrm{cannot}\:\mathrm{work}\:\mathrm{sir},\:\:\mathrm{and}\:\mathrm{please}\:\mathrm{show}\:\mathrm{me}\:\mathrm{the}\:\:\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} } } \:\:=\:\:\mathrm{729} \\ $$
Commented by mr W last updated on 02/May/19
![(x^x )^x^x =a let t=x^x ⇒t^t =a ⇒t=((ln a)/(W(ln a))) ⇒x^x =((ln a)/(W(ln a)))=b ⇒x=((ln b)/(W(ln b)))=((ln [((ln a)/(W(ln a)))])/(W{ln [((ln a)/(W(ln a)))]}))](Q58908.png)
$$\left({x}^{{x}} \right)^{{x}^{{x}} } ={a} \\ $$$${let}\:{t}={x}^{{x}} \\ $$$$\Rightarrow{t}^{{t}} ={a} \\ $$$$\Rightarrow{t}=\frac{\mathrm{ln}\:{a}}{{W}\left(\mathrm{ln}\:{a}\right)} \\ $$$$\Rightarrow{x}^{{x}} =\frac{\mathrm{ln}\:{a}}{{W}\left(\mathrm{ln}\:{a}\right)}={b} \\ $$$$\Rightarrow{x}=\frac{\mathrm{ln}\:{b}}{{W}\left(\mathrm{ln}\:{b}\right)}=\frac{\mathrm{ln}\:\left[\frac{\mathrm{ln}\:{a}}{{W}\left(\mathrm{ln}\:{a}\right)}\right]}{{W}\left\{\mathrm{ln}\:\left[\frac{\mathrm{ln}\:{a}}{{W}\left(\mathrm{ln}\:{a}\right)}\right]\right\}} \\ $$
Commented by Tawa1 last updated on 01/May/19
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Answered by MJS last updated on 01/May/19
$${x}^{{x}^{{x}} } ={x}^{\left({x}^{{x}} \right)} =\mathrm{729} \\ $$$$\mathrm{approximation}\:\mathrm{leads}\:\mathrm{to} \\ $$$${x}\approx\mathrm{2}.\mathrm{364994} \\ $$$$\left({x}^{{x}} \right)^{{x}} ={x}^{\left({x}^{\mathrm{2}} \right)} =\mathrm{729}\:\Rightarrow\:{x}\approx\mathrm{2}.\mathrm{617397} \\ $$
Commented by Tawa1 last updated on 01/May/19
$$\mathrm{But}\:\mathrm{different}\:\mathrm{answer}\:\mathrm{sir}.\:\:\:\:\:\mathrm{x}^{\mathrm{x}^{\mathrm{x}} \:} \:\neq\:\:\mathrm{x}^{\mathrm{x}^{\mathrm{2}} } \\ $$
Commented by Tawa1 last updated on 01/May/19
$$\mathrm{And}\:\mathrm{any}\:\mathrm{workings}\:\mathrm{sir} \\ $$
Commented by MJS last updated on 01/May/19
$$\mathrm{I}\:\mathrm{know}\:{x}^{{x}^{{x}} } \neq{x}^{{x}^{\mathrm{2}} } \:\mathrm{but}\:\mathrm{I}\:\mathrm{wasn}'\mathrm{t}\:\mathrm{100\%}\:\mathrm{sure}\:\mathrm{if} \\ $$$$\mathrm{you}\:\mathrm{know}\:\mathrm{this}\:\mathrm{too}\:\mathrm{because}\:\mathrm{it}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{seem} \\ $$$$\mathrm{to}\:\mathrm{be}\:\mathrm{common}\:\mathrm{knowledge}. \\ $$$$\mathrm{I}\:\mathrm{approximated}\:\mathrm{with}\:\mathrm{an}\:\mathrm{electronic}\:\mathrm{calculator} \\ $$$$\mathrm{so}\:\mathrm{I}\:\mathrm{can}'\mathrm{t}\:\mathrm{show}\:\mathrm{any}\:\mathrm{workings} \\ $$$$\mathrm{I}\:\mathrm{calculated}\:{f}\left({x}\right)={x}^{{x}^{{x}} } −\mathrm{729}\:\mathrm{starting}\:\mathrm{with} \\ $$$${f}\left(\mathrm{2}\right)=−\mathrm{713} \\ $$$${f}\left(\mathrm{3}\right)=\mathrm{7}.\mathrm{62}×\mathrm{10}^{\mathrm{12}} \\ $$$${f}\left(\mathrm{2}.\mathrm{5}\right)=\mathrm{7831}.\mathrm{31}... \\ $$$${f}\left(\mathrm{2}.\mathrm{1}\right)=−\mathrm{695}.\mathrm{08}... \\ $$$${f}\left(\mathrm{2}.\mathrm{2}\right)=−\mathrm{641}.\mathrm{82}... \\ $$$${f}\left(\mathrm{2}.\mathrm{3}\right)=−\mathrm{442}.\mathrm{76}... \\ $$$${f}\left(\mathrm{2}.\mathrm{4}\right)=\mathrm{554}.\mathrm{40}... \\ $$$${f}\left(\mathrm{2}.\mathrm{35}\right)=−\mathrm{148}.\mathrm{87}... \\ $$$${f}\left(\mathrm{2}.\mathrm{36}\right)=−\mathrm{54}.\mathrm{00}... \\ $$$${f}\left(\mathrm{2}.\mathrm{37}\right)=\mathrm{59}.\mathrm{19}... \\ $$$${f}\left(\mathrm{2}.\mathrm{365}\right)=.\mathrm{067773}... \\ $$$${f}\left(\mathrm{2}.\mathrm{364}\right)=−\mathrm{11}.\mathrm{13}... \\ $$$${f}\left(\mathrm{2}.\mathrm{3645}\right)=−\mathrm{5}.\mathrm{55}... \\ $$$$\mathrm{and}\:\mathrm{so}\:\mathrm{on} \\ $$