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Question Number 109193 by mr W last updated on 21/Aug/20

$$solve$$$$x^{x^3}=5$$

Commented by aurpeyz last updated on 21/Aug/20

$$x^{x^3}=2^2$$$$x^3=2$$$$x{\underset{}{=}}^3\sqrt{2}$$

Commented by aurpeyz last updated on 21/Aug/20

$$ijusttried.plscorrectme$$

Commented by mr W last updated on 21/Aug/20

$$if{x}^3=2thenx=2,impossble.$$

Commented by mr W last updated on 21/Aug/20

$$ihavechangedthequestionto$$$$x^{x^3}=5$$

Commented by aurpeyz last updated on 21/Aug/20

$$iwrotecuberootof2$$

Commented by aurpeyz last updated on 21/Aug/20

$$okaysir.iwilltry$$

Commented by mr W last updated on 21/Aug/20

$$youwrote$$$$x^{x^3}=2^2$$$$if{x}^3=2,thenyoumeanalsox=2.$$

Commented by aurpeyz last updated on 21/Aug/20

$$yes.iamwrong.iftheindecesareequal.thebase$$$$willbeequal.$$

Answered by Dwaipayan Shikari last updated on 21/Aug/20

$$x^{3}logx=log5$$$$3logx{e}^{3logx}=3log5$$$$3logx=W_0\left(3log5\right)$$$$x=e^{\frac{W_0\left(3log5\right)}{3}}=1.5459..$$

Commented by mr W last updated on 21/Aug/20

$$good!$$

Answered by Aziztisffola last updated on 21/Aug/20

$$\ln \left({\mathrm{x}}^{{\mathrm{x}}^3}\right)=\ln \left(5\right)\iff {\mathrm{x}}^3\ln \left(\mathrm{x}\right)=\ln \left(5\right)$$$$\iff \ln \left(\mathrm{x}\right){\mathrm{e}}^{3\ln \left(\mathrm{x}\right)}=\ln \left(5\right)$$$$\iff 3\ln \left(\mathrm{x}\right){\mathrm{e}}^{3\ln \left(\mathrm{x}\right)}=3\ln \left(5\right)$$$$\mathrm{W}\left(3\ln \left(\mathrm{x}\right){\mathrm{e}}^{3\ln \left(\mathrm{x}\right)}\right)=\mathrm{W}\left(3\ln \left(5\right)\right)$$$$3\ln \left(\mathrm{x}\right)=\mathrm{W}\left(3\ln \left(5\right)\right)$$$$\mathrm{x}={\mathrm{e}}^{\frac{\mathrm{W}\left(3\ln \left(5\right)\right)}{3}}$$

Commented by aurpeyz last updated on 24/Aug/20

$$\mathrm{Pls}\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{w}\mathrm{or}{\mathrm{w}}_0\mathrm{being}\mathrm{inserted}?$$

Commented by Aziztisffola last updated on 24/Aug/20

$$\mathrm{The}\mathrm{Lambert}\mathrm{W}\mathrm{function};\mathrm{the}\mathrm{invers}$$$$\mathrm{function}\mathrm{of}f:x\mid \to {xe}^x$$$$\mathrm{W}=f^{-1}$$

Commented by aurpeyz last updated on 25/Aug/20

$$wow.thatwaswhyyoumultipliedthrough$$$$by3?$$

Answered by 1549442205PVT last updated on 22/Aug/20

$$\mathrm{Put}{\mathrm{x}}^3=\mathrm{y}\Rightarrow \mathrm{x}={}^3\sqrt{\mathrm{y}}\Rightarrow {}^3{\left(\sqrt{\mathrm{y}}\right)}^{\mathrm{y}}=5$$$$\mathrm{yln}\left({}^3\sqrt{\mathrm{y}}\right)=\ln 5\iff \frac{\mathrm{y}}{3}\mathrm{lny}=\ln 5$$$$\mathrm{ylny}=3\ln 5\iff \mathrm{y}={\mathrm{e}}^{\frac{3\ln 5}{\mathrm{y}}}\iff \frac{1}{\mathrm{y}}.{\mathrm{e}}^{\frac{3\ln 5}{\mathrm{y}}}=1$$$$\mathrm{Put}\frac{1}{\mathrm{y}}=\mathrm{z}\mathrm{we}\mathrm{get}{\mathrm{ze}}^{\mathrm{z}.3\ln 5}=1$$$$\iff \mathrm{z}.{3\ln 5\ln 5\mathrm{e}}^{\mathrm{z}.3\ln 5}=3\ln 5$$$$\Rightarrow \mathrm{z}.3\ln 5={\mathrm{W}}_0\left(3\ln 5\right)=1.306568..$$$$\Rightarrow \mathrm{z}=1.306568/\left(3\ln 5\right)=0.27060.$$$$\Rightarrow \mathrm{y}=1/\mathrm{z}=3.695417$$$$\Rightarrow \mathrm{x}={}^3\sqrt{\mathrm{y}}=1.5460$$

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