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Question Number 6846 by Tawakalitu. last updated on 30/Jul/16

x^x  = 16    find the value of x

$${x}^{{x}} \:=\:\mathrm{16} \\ $$$$ \\ $$$${find}\:{the}\:{value}\:{of}\:{x}\: \\ $$

Commented by Rasheed Soomro last updated on 31/Jul/16

  2^2 <16<3^3   So,     2<x<3  Can we determine x by interpolation?  Question for the sake of learning.

$$\:\:\mathrm{2}^{\mathrm{2}} <\mathrm{16}<\mathrm{3}^{\mathrm{3}} \\ $$$${So},\:\:\:\:\:\mathrm{2}<{x}<\mathrm{3} \\ $$$${Can}\:{we}\:{determine}\:{x}\:{by}\:\boldsymbol{{interpolation}}? \\ $$$${Question}\:{for}\:{the}\:{sake}\:{of}\:{learning}. \\ $$

Commented by Tawakalitu. last updated on 31/Jul/16

Please use the lambert function

$${Please}\:{use}\:{the}\:{lambert}\:{function} \\ $$

Commented by prakash jain last updated on 31/Jul/16

xln x=ln 16  e^(ln x) ln x=ln 16  W(e^(ln x) ln x)=W(ln 16)  ln x=W(ln 16)  x=e^(W(ln 16))

$${x}\mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{16} \\ $$$${e}^{\mathrm{ln}\:{x}} \mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{16} \\ $$$${W}\left({e}^{\mathrm{ln}\:{x}} \mathrm{ln}\:{x}\right)={W}\left(\mathrm{ln}\:\mathrm{16}\right) \\ $$$$\mathrm{ln}\:{x}={W}\left(\mathrm{ln}\:\mathrm{16}\right) \\ $$$${x}={e}^{{W}\left(\mathrm{ln}\:\mathrm{16}\right)} \\ $$

Commented by Tawakalitu. last updated on 31/Jul/16

Thanks so much

$${Thanks}\:{so}\:{much} \\ $$

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