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Question Number 45314 by mondodotto@gmail.com last updated on 11/Oct/18
solve for x 10^x =x^(50)
$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{{x}} \\ $$$$\mathrm{10}^{\boldsymbol{{x}}} =\boldsymbol{{x}}^{\mathrm{50}} \\ $$
Answered by MrW3 last updated on 12/Oct/18
10^x =x^(50) ±10^(x/(50)) =x ±e^((x/(50))ln 10) =x xe^(−(x/(50))ln 10) =±1 −((xln 10)/(50))e^(−(x/(50))ln 10) =∓((ln 10)/(50)) ⇒−((xln 10)/(50))=W(∓((ln 10)/(50))) ⇒x=−((50)/(ln 10))W(∓((ln 10)/(50))) = { ((−((50)/(ln 10))W(((ln 10)/(50)))=−((50×0.044066)/(ln 10))=−0.956881)),((−((50)/(ln 10))W(−((ln 10)/(50)))= { ((−((50×(−0.0483321))/(ln 10))=1.049518)),((−((50×(−4.605170))/(ln 10))=100)) :})) :}
$$\mathrm{10}^{{x}} ={x}^{\mathrm{50}} \\ $$$$\pm\mathrm{10}^{\frac{{x}}{\mathrm{50}}} ={x} \\ $$$$\pm{e}^{\frac{{x}}{\mathrm{50}}\mathrm{ln}\:\mathrm{10}} ={x} \\ $$$${xe}^{−\frac{{x}}{\mathrm{50}}\mathrm{ln}\:\mathrm{10}} =\pm\mathrm{1} \\ $$$$−\frac{{x}\mathrm{ln}\:\mathrm{10}}{\mathrm{50}}{e}^{−\frac{{x}}{\mathrm{50}}\mathrm{ln}\:\mathrm{10}} =\mp\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{50}} \\ $$$$\Rightarrow−\frac{{x}\mathrm{ln}\:\mathrm{10}}{\mathrm{50}}={W}\left(\mp\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{50}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{50}}{\mathrm{ln}\:\mathrm{10}}{W}\left(\mp\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{50}}\right) \\ $$$$=\begin{cases}{−\frac{\mathrm{50}}{\mathrm{ln}\:\mathrm{10}}{W}\left(\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{50}}\right)=−\frac{\mathrm{50}×\mathrm{0}.\mathrm{044066}}{\mathrm{ln}\:\mathrm{10}}=−\mathrm{0}.\mathrm{956881}}\\{−\frac{\mathrm{50}}{\mathrm{ln}\:\mathrm{10}}{W}\left(−\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{50}}\right)=\begin{cases}{−\frac{\mathrm{50}×\left(−\mathrm{0}.\mathrm{0483321}\right)}{\mathrm{ln}\:\mathrm{10}}=\mathrm{1}.\mathrm{049518}}\\{−\frac{\mathrm{50}×\left(−\mathrm{4}.\mathrm{605170}\right)}{\mathrm{ln}\:\mathrm{10}}=\mathrm{100}}\end{cases}}\end{cases} \\ $$
Commented by mondodotto@gmail.com last updated on 12/Oct/18
thanx atleast i get something
$$\mathrm{thanx}\:\mathrm{atleast}\:\mathrm{i}\:\mathrm{get}\:\mathrm{something} \\ $$
Commented by mondodotto@gmail.com last updated on 12/Oct/18
thanx a lot but i dont undstnd this method please can you explain sir
$$\mathrm{thanx}\:\mathrm{a}\:\mathrm{lot}\:\mathrm{but}\:\mathrm{i}\:\mathrm{dont}\:\boldsymbol{\mathrm{undstnd}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{method}}\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{can}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{explain}}\:\boldsymbol{\mathrm{sir}} \\ $$
Commented by MrW3 last updated on 12/Oct/18
Lambert W function is used. If xe^x =a then x=W(a).
$${Lambert}\:{W}\:{function}\:{is}\:{used}. \\ $$$${If}\:{xe}^{{x}} ={a}\:{then}\:{x}={W}\left({a}\right). \\ $$
Commented by mondodotto@gmail.com last updated on 12/Oct/18
what is W
$$\mathrm{what}\:\mathrm{is}\:\boldsymbol{{W}} \\ $$
Commented by MrW3 last updated on 12/Oct/18
To apply Lambert function to solve the equation, all what we have to do is to put the equation into a form like Xe^X =Y then we get X=W(Y).
$${To}\:{apply}\:{Lambert}\:{function}\:{to}\:{solve} \\ $$$${the}\:{equation},\:{all}\:{what}\:{we}\:{have}\:{to}\:{do} \\ $$$${is}\:{to}\:{put}\:{the}\:{equation}\:{into}\:{a}\:{form}\:{like} \\ $$$${Xe}^{{X}} ={Y} \\ $$$${then}\:{we}\:{get}\:{X}=\mathbb{W}\left({Y}\right). \\ $$
Commented by MrW3 last updated on 12/Oct/18
W(x) is the lambert function, defined as W(x)e^(W(x)) =x.
$${W}\left({x}\right)\:{is}\:{the}\:{lambert}\:{function},\:{defined} \\ $$$${as}\:{W}\left({x}\right){e}^{{W}\left({x}\right)} ={x}. \\ $$

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