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Question-201192




Question Number 201192 by Euclid last updated on 01/Dec/23
Answered by Calculusboy last updated on 01/Dec/23
Solution: Apply In to both sides  In x^x^4  =In64  x^4 Inx=In64  Inx∙e^(4Inx) =In64     [Nb: x^4 =e^(4Inx) ]  multiply through by 4  4Inx∙e^(4Inx) =4In64  apply lambert function to both sides  W[4Inx∙e^(4Inx) ]=W[4In2^6 ]  W[4Inx∙e^(4Inx) ]=W[4∙6In2]    ⇔  W[24In2]  W[4Inx∙e^(4Inx) ]=W[8∙3In2]  ⇔W[8In2^3 ]  W[4Inx∙e^(4Inx) ]=W[8In8]  Nb: W(xe^x )=x  and W(aIna)=Ina  4Inx=In8    Inx^4 =In8  x^4 =8  x=(8)^(1/4)
$$\boldsymbol{{Solution}}:\:\boldsymbol{{Apply}}\:\boldsymbol{{I}\mathrm{n}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{{both}}\:\boldsymbol{{sides}} \\ $$$$\boldsymbol{{In}}\:\boldsymbol{{x}}^{\boldsymbol{{x}}^{\mathrm{4}} } =\boldsymbol{{In}}\mathrm{64} \\ $$$$\boldsymbol{{x}}^{\mathrm{4}} \boldsymbol{{Inx}}=\boldsymbol{{In}}\mathrm{64} \\ $$$$\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} =\boldsymbol{{In}}\mathrm{64}\:\:\:\:\:\left[\boldsymbol{{Nb}}:\:\boldsymbol{{x}}^{\mathrm{4}} =\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right] \\ $$$$\boldsymbol{{multiply}}\:\boldsymbol{{through}}\:\boldsymbol{{by}}\:\mathrm{4} \\ $$$$\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} =\mathrm{4}\boldsymbol{{In}}\mathrm{64} \\ $$$$\boldsymbol{{apply}}\:\boldsymbol{{lambert}}\:\boldsymbol{{function}}\:\boldsymbol{{to}}\:\boldsymbol{{both}}\:\boldsymbol{{sides}} \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{In}}\mathrm{2}^{\mathrm{6}} \right] \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{4}\centerdot\mathrm{6}\boldsymbol{{In}}\mathrm{2}\right]\:\:\:\:\Leftrightarrow\:\:\boldsymbol{{W}}\left[\mathrm{24}\boldsymbol{{In}}\mathrm{2}\right] \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{8}\centerdot\mathrm{3}\boldsymbol{{In}}\mathrm{2}\right]\:\:\Leftrightarrow\boldsymbol{{W}}\left[\mathrm{8}\boldsymbol{{In}}\mathrm{2}^{\mathrm{3}} \right] \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{8}\boldsymbol{{In}}\mathrm{8}\right] \\ $$$$\boldsymbol{{Nb}}:\:\boldsymbol{{W}}\left(\boldsymbol{{xe}}^{\boldsymbol{{x}}} \right)=\boldsymbol{{x}}\:\:\boldsymbol{{and}}\:\boldsymbol{{W}}\left(\boldsymbol{{aIna}}\right)=\boldsymbol{{Ina}} \\ $$$$\mathrm{4}\boldsymbol{{Inx}}=\boldsymbol{{In}}\mathrm{8}\:\: \\ $$$$\boldsymbol{{Inx}}^{\mathrm{4}} =\boldsymbol{{In}}\mathrm{8} \\ $$$$\boldsymbol{{x}}^{\mathrm{4}} =\mathrm{8} \\ $$$$\boldsymbol{{x}}=\sqrt[{\mathrm{4}}]{\mathrm{8}} \\ $$$$ \\ $$

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