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Question-12365

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Question Number 12365 by chux last updated on 20/Apr/17
Commented by mrW1 last updated on 21/Apr/17
Definition of Lambert W function: if y=xe^x then x=W(y) i.e. y=W(y)e^(W(y)) All what you have to do is to transform your equation into the form like this Y=Xe^X then you can use the W−function to get X=W(Y) Sometimes it′s usefull to know a^b =e^(ln a^b ) =e^(bln a) For example: solve log x^2 =(x/8) ⇒x^2 =10^(((x/8))) ⇒x=±10^(((x/(16)))) ⇒x=±e^(((x/(16)))ln 10) ⇒x×e^(−((ln 10)/(16))x) =±1 ⇒(−((ln 10)/(16))x)×e^((−((ln 10)/(16))x)) =±((ln 10)/(16)) ⇒−((ln 10)/(16))x=W(±((ln 10)/(16))) ⇒x=−((W(±((ln 10)/(16))))/((ln 10)/(16))) ≈−((W(±0.143911))/(0.143911)) = { ((−((W(0.143911))/(0.143911))=−((0.126776)/(0.143911))=−0.88093)),((−((W(−0.143911))/(0.143911))= { ((−((−0.170697)/(0.143911))=1.18613)),((−((−3.05550)/(0.143911))=21.23178)) :})) :} For example: solve x^x =16 ⇒xln x=ln 16 ⇒(ln x)e^((ln x)) =ln 16 ⇒ln x=W(ln 16) ⇒x=e^(W(ln 16)) =((ln 16)/(W(ln 16))) ≈((2.771588)/(W(2.771588)))=((2.771588)/(1.00973))=2.74587 In internet you may find calculators to evaluate W function values.
$${Definition}\:{of}\:{Lambert}\:{W}\:{function}: \\ $$$${if}\:{y}={xe}^{{x}} \\ $$$${then}\:{x}={W}\left({y}\right) \\ $$$${i}.{e}.\:{y}={W}\left({y}\right){e}^{{W}\left({y}\right)} \\ $$$$ \\ $$$${All}\:{what}\:{you}\:{have}\:{to}\:{do}\:{is}\:{to}\:{transform} \\ $$$${your}\:{equation}\:{into}\:{the}\:{form}\:{like}\:{this} \\ $$$${Y}={X}\boldsymbol{{e}}^{{X}} \\ $$$${then}\:{you}\:{can}\:{use}\:{the}\:{W}−{function}\:{to}\:{get} \\ $$$${X}={W}\left({Y}\right) \\ $$$$ \\ $$$${Sometimes}\:{it}'{s}\:{usefull}\:{to}\:{know}\: \\ $$$${a}^{{b}} ={e}^{\mathrm{ln}\:{a}^{{b}} } ={e}^{{b}\mathrm{ln}\:{a}} \\ $$$$ \\ $$$${For}\:{example}:\:{solve}\:\mathrm{log}\:{x}^{\mathrm{2}} =\frac{{x}}{\mathrm{8}} \\ $$$$\Rightarrow{x}^{\mathrm{2}} =\mathrm{10}^{\left(\frac{{x}}{\mathrm{8}}\right)} \\ $$$$\Rightarrow{x}=\pm\mathrm{10}^{\left(\frac{{x}}{\mathrm{16}}\right)} \\ $$$$\Rightarrow{x}=\pm{e}^{\left(\frac{{x}}{\mathrm{16}}\right)\mathrm{ln}\:\mathrm{10}} \\ $$$$\Rightarrow{x}×{e}^{−\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}}{x}} =\pm\mathrm{1} \\ $$$$\Rightarrow\left(−\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}}{x}\right)×{e}^{\left(−\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}}{x}\right)} =\pm\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}} \\ $$$$\Rightarrow−\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}}{x}={W}\left(\pm\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}}\right) \\ $$$$\Rightarrow{x}=−\frac{{W}\left(\pm\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}}\right)}{\frac{\mathrm{ln}\:\mathrm{10}}{\mathrm{16}}} \\ $$$$\approx−\frac{{W}\left(\pm\mathrm{0}.\mathrm{143911}\right)}{\mathrm{0}.\mathrm{143911}} \\ $$$$=\begin{cases}{−\frac{{W}\left(\mathrm{0}.\mathrm{143911}\right)}{\mathrm{0}.\mathrm{143911}}=−\frac{\mathrm{0}.\mathrm{126776}}{\mathrm{0}.\mathrm{143911}}=−\mathrm{0}.\mathrm{88093}}\\{−\frac{{W}\left(−\mathrm{0}.\mathrm{143911}\right)}{\mathrm{0}.\mathrm{143911}}=\begin{cases}{−\frac{−\mathrm{0}.\mathrm{170697}}{\mathrm{0}.\mathrm{143911}}=\mathrm{1}.\mathrm{18613}}\\{−\frac{−\mathrm{3}.\mathrm{05550}}{\mathrm{0}.\mathrm{143911}}=\mathrm{21}.\mathrm{23178}}\end{cases}}\end{cases} \\ $$$$ \\ $$$${For}\:{example}:\:{solve}\:{x}^{{x}} =\mathrm{16} \\ $$$$\Rightarrow{x}\mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{16} \\ $$$$\Rightarrow\left(\mathrm{ln}\:{x}\right){e}^{\left(\mathrm{ln}\:{x}\right)} =\mathrm{ln}\:\mathrm{16} \\ $$$$\Rightarrow\mathrm{ln}\:{x}={W}\left(\mathrm{ln}\:\mathrm{16}\right) \\ $$$$\Rightarrow{x}={e}^{{W}\left(\mathrm{ln}\:\mathrm{16}\right)} =\frac{\mathrm{ln}\:\mathrm{16}}{{W}\left(\mathrm{ln}\:\mathrm{16}\right)} \\ $$$$\approx\frac{\mathrm{2}.\mathrm{771588}}{{W}\left(\mathrm{2}.\mathrm{771588}\right)}=\frac{\mathrm{2}.\mathrm{771588}}{\mathrm{1}.\mathrm{00973}}=\mathrm{2}.\mathrm{74587} \\ $$$$ \\ $$$${In}\:{internet}\:{you}\:{may}\:{find}\:{calculators} \\ $$$${to}\:{evaluate}\:{W}\:{function}\:{values}. \\ $$
Commented by chux last updated on 20/Apr/17
i ve always had problems with the final simplification using LAMBART W FUNCTION . is it that there s a special calculator for it or what. please help me out if theres any calculator or rule for it.Also help with its properties or rules. Thanks for always helping.
$$\mathrm{i}\:\mathrm{ve}\:\mathrm{always}\:\mathrm{had}\:\mathrm{problems}\:\mathrm{with}\:\mathrm{the} \\ $$$$\mathrm{final}\:\mathrm{simplification}\:\mathrm{using}\: \\ $$$$\mathrm{LAMBART}\:\mathrm{W}\:\mathrm{FUNCTION}\:. \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{that}\:\mathrm{there}\:\mathrm{s}\:\mathrm{a}\:\mathrm{special}\:\mathrm{calculator} \\ $$$$\mathrm{for}\:\mathrm{it}\:\mathrm{or}\:\mathrm{what}.\: \\ $$$$ \\ $$$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{out}\:\mathrm{if}\:\mathrm{theres}\:\mathrm{any}\:\: \\ $$$$\mathrm{calculator}\:\mathrm{or}\:\mathrm{rule}\:\mathrm{for}\:\mathrm{it}.\mathrm{Also}\:\mathrm{help}\: \\ $$$$\mathrm{with}\:\mathrm{its}\:\mathrm{properties}\:\mathrm{or}\:\mathrm{rules}. \\ $$$$ \\ $$$$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{always}\:\mathrm{helping}. \\ $$
Commented by chux last updated on 21/Apr/17
can you recomend any calculator for it.
$$\mathrm{can}\:\mathrm{you}\:\mathrm{recomend}\:\mathrm{any}\:\mathrm{calculator}\: \\ $$$$\mathrm{for}\:\mathrm{it}. \\ $$
Commented by mrW1 last updated on 21/Apr/17
There was a very nice online calculator at www.had2know.com, but the site is no longer valid. I use geogebra to evaluate W(a) indirectly.
$${There}\:{was}\:{a}\:{very}\:{nice}\:{online}\: \\ $$$${calculator}\:{at}\:{www}.{had}\mathrm{2}{know}.{com}, \\ $$$${but}\:{the}\:{site}\:{is}\:{no}\:{longer}\:{valid}. \\ $$$${I}\:{use}\:{geogebra}\:{to}\:{evaluate}\:{W}\left({a}\right) \\ $$$${indirectly}. \\ $$
Commented by chux last updated on 21/Apr/17
thank you very much sir
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{sir} \\ $$

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