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Question Number 16906 by tawa tawa last updated on 28/Jun/17

x^x = 100, find x.

$$\mathrm{x}^{\mathrm{x}} \:=\:\mathrm{100},\:\mathrm{find}\:\mathrm{x}. \\ $$

Answered by mrW1 last updated on 28/Jun/17

x^x = 100 x=100^(1/x) x=e^((1/x)ln 100) (1/x)e^((1/x)ln 100) =1 ((1/x)ln 100)e^((1/x)ln 100) =ln 100 ⇒(1/x)ln 100=W(ln 100) ⇒x=((ln 100)/(W(ln 100)))≈((4.60517)/(W(4.60517))) ≈((4.60517)/(1.28018))=3.59728 Generally: solution for x^x =a is x=((ln a)/(W(ln a)))

$$\mathrm{x}^{\mathrm{x}} \:=\:\mathrm{100} \\ $$$$\mathrm{x}=\mathrm{100}^{\frac{\mathrm{1}}{\mathrm{x}}} \\ $$$$\mathrm{x}=\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}\mathrm{ln}\:\mathrm{100}} \\ $$$$\frac{\mathrm{1}}{\mathrm{x}}\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}\mathrm{ln}\:\mathrm{100}} =\mathrm{1} \\ $$$$\left(\frac{\mathrm{1}}{\mathrm{x}}\mathrm{ln}\:\mathrm{100}\right)\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}\mathrm{ln}\:\mathrm{100}} =\mathrm{ln}\:\mathrm{100} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{x}}\mathrm{ln}\:\mathrm{100}=\mathrm{W}\left(\mathrm{ln}\:\mathrm{100}\right) \\ $$$$\Rightarrow\mathrm{x}=\frac{\mathrm{ln}\:\mathrm{100}}{\mathrm{W}\left(\mathrm{ln}\:\mathrm{100}\right)}\approx\frac{\mathrm{4}.\mathrm{60517}}{\mathrm{W}\left(\mathrm{4}.\mathrm{60517}\right)} \\ $$$$\approx\frac{\mathrm{4}.\mathrm{60517}}{\mathrm{1}.\mathrm{28018}}=\mathrm{3}.\mathrm{59728} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Generally}: \\ $$$$\mathrm{solution}\:\mathrm{for}\:\mathrm{x}^{\mathrm{x}} =\mathrm{a}\:\mathrm{is} \\ $$$$\mathrm{x}=\frac{\mathrm{ln}\:\mathrm{a}}{\mathrm{W}\left(\mathrm{ln}\:\mathrm{a}\right)} \\ $$

Commented by tawa tawa last updated on 28/Jun/17

God bless you sir. i really appreciate.

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

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