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Question Number 9392 by tawakalitu last updated on 04/Dec/16

x^x  = 16  find the value of x.  please show workings.

$$\mathrm{x}^{\mathrm{x}} \:=\:\mathrm{16} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}. \\ $$$$\mathrm{please}\:\mathrm{show}\:\mathrm{workings}. \\ $$

Answered by mrW last updated on 04/Dec/16

using Lambert W function  x^x =16  ⇒ xln x=ln 16  let x=e^u   ue^u =ln 16  ⇒ u=W(ln 16)  ⇒ x=e^(W(ln 16)) =((ln 16)/(W(ln 16)))  ≈((2.77258872223978)/(W(2.77258872223978)))  ≈((2.77258872223978)/(1.0099151364912))  ≈2.74536802356749

$$\mathrm{using}\:\mathrm{Lambert}\:\boldsymbol{\mathrm{W}}\:\mathrm{function} \\ $$$$\mathrm{x}^{\mathrm{x}} =\mathrm{16} \\ $$$$\Rightarrow\:\mathrm{xln}\:\mathrm{x}=\mathrm{ln}\:\mathrm{16} \\ $$$$\mathrm{let}\:\mathrm{x}=\mathrm{e}^{\mathrm{u}} \\ $$$$\mathrm{ue}^{\mathrm{u}} =\mathrm{ln}\:\mathrm{16} \\ $$$$\Rightarrow\:\mathrm{u}=\mathrm{W}\left(\mathrm{ln}\:\mathrm{16}\right) \\ $$$$\Rightarrow\:\mathrm{x}=\mathrm{e}^{\mathrm{W}\left(\mathrm{ln}\:\mathrm{16}\right)} =\frac{\mathrm{ln}\:\mathrm{16}}{\mathrm{W}\left(\mathrm{ln}\:\mathrm{16}\right)} \\ $$$$\approx\frac{\mathrm{2}.\mathrm{77258872223978}}{\mathrm{W}\left(\mathrm{2}.\mathrm{77258872223978}\right)} \\ $$$$\approx\frac{\mathrm{2}.\mathrm{77258872223978}}{\mathrm{1}.\mathrm{0099151364912}} \\ $$$$\approx\mathrm{2}.\mathrm{74536802356749} \\ $$

Commented by tawakalitu last updated on 04/Dec/16

i really appreciate sir. God bless you.

$$\mathrm{i}\:\mathrm{really}\:\mathrm{appreciate}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

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