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)](https://www.tinkutara.com/question/Q201195.png)
$$\boldsymbol{{Solution}}:\:\boldsymbol{{Apply}}\:\boldsymbol{{I}\mathrm{n}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{{both}}\:\boldsymbol{{sides}} \\ $$$$\boldsymbol{{In}}\:\boldsymbol{{x}}^{\boldsymbol{{x}}^{\mathrm{4}} } =\boldsymbol{{In}}\mathrm{64} \\ $$$$\boldsymbol{{x}}^{\mathrm{4}} \boldsymbol{{Inx}}=\boldsymbol{{In}}\mathrm{64} \\ $$$$\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} =\boldsymbol{{In}}\mathrm{64}\:\:\:\:\:\left[\boldsymbol{{Nb}}:\:\boldsymbol{{x}}^{\mathrm{4}} =\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right] \\ $$$$\boldsymbol{{multiply}}\:\boldsymbol{{through}}\:\boldsymbol{{by}}\:\mathrm{4} \\ $$$$\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} =\mathrm{4}\boldsymbol{{In}}\mathrm{64} \\ $$$$\boldsymbol{{apply}}\:\boldsymbol{{lambert}}\:\boldsymbol{{function}}\:\boldsymbol{{to}}\:\boldsymbol{{both}}\:\boldsymbol{{sides}} \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{In}}\mathrm{2}^{\mathrm{6}} \right] \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{4}\centerdot\mathrm{6}\boldsymbol{{In}}\mathrm{2}\right]\:\:\:\:\Leftrightarrow\:\:\boldsymbol{{W}}\left[\mathrm{24}\boldsymbol{{In}}\mathrm{2}\right] \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{8}\centerdot\mathrm{3}\boldsymbol{{In}}\mathrm{2}\right]\:\:\Leftrightarrow\boldsymbol{{W}}\left[\mathrm{8}\boldsymbol{{In}}\mathrm{2}^{\mathrm{3}} \right] \\ $$$$\boldsymbol{{W}}\left[\mathrm{4}\boldsymbol{{Inx}}\centerdot\boldsymbol{{e}}^{\mathrm{4}\boldsymbol{{Inx}}} \right]=\boldsymbol{{W}}\left[\mathrm{8}\boldsymbol{{In}}\mathrm{8}\right] \\ $$$$\boldsymbol{{Nb}}:\:\boldsymbol{{W}}\left(\boldsymbol{{xe}}^{\boldsymbol{{x}}} \right)=\boldsymbol{{x}}\:\:\boldsymbol{{and}}\:\boldsymbol{{W}}\left(\boldsymbol{{aIna}}\right)=\boldsymbol{{Ina}} \\ $$$$\mathrm{4}\boldsymbol{{Inx}}=\boldsymbol{{In}}\mathrm{8}\:\: \\ $$$$\boldsymbol{{Inx}}^{\mathrm{4}} =\boldsymbol{{In}}\mathrm{8} \\ $$$$\boldsymbol{{x}}^{\mathrm{4}} =\mathrm{8} \\ $$$$\boldsymbol{{x}}=\sqrt[{\mathrm{4}}]{\mathrm{8}} \\ $$$$ \\ $$