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Question Number 185076 by mathlove last updated on 16/Jan/23

$$x^{2x^6}=3$$$$x=?$$

Answered by aba last updated on 16/Jan/23

$${\mathrm{x}}^{{2\mathrm{x}}^6}=3\Rightarrow {2\mathrm{x}}^6\ln \left(\mathrm{x}\right)=\ln \left(3\right)$$$$\Rightarrow {\mathrm{x}}^6\ln \left(\mathrm{x}\right)=\frac{\ln \left(3\right)}{2}$$$$\Rightarrow \ln \left(\mathrm{x}\right)\times {\mathrm{e}}^{\ln \left({\mathrm{x}}^6\right)}=\frac{\ln \left(3\right)}{2}$$$$\Rightarrow 6\ln \left(\mathrm{x}\right)\times {\mathrm{e}}^{\ln \left({\mathrm{x}}^6\right)}=\frac{6\ln \left(3\right)}{2}$$$$\Rightarrow \mathrm{W}\left(\ln \left({\mathrm{x}}^6\right)\times {\mathrm{e}}^{\ln \left({\mathrm{x}}^6\right)}\right)=\mathrm{W}\left(3\ln \left(3\right)\right)$$$$\Rightarrow \ln \left({\mathrm{x}}^6\right)=\mathrm{W}\left(3\ln \left(3\right)\right)$$$$\Rightarrow {\mathrm{x}}^6={\mathrm{e}}^{\mathrm{W}\left(3\ln \left(3\right)\right)}$$$$\Rightarrow {\mathrm{x}}^6=\frac{3\ln \left(3\right)}{\mathrm{W}\left(3\ln \left(3\right)\right)}$$$$\Rightarrow {\mathrm{x}}^6=\frac{3\ln \left(3\right)}{\mathrm{W}\left(\ln \left(3\right){\mathrm{e}}^{\ln \left(3\right)}\right)}$$$$\Rightarrow {\mathrm{x}}^6=\frac{3\ln \left(3\right)}{\ln \left(3\right)}$$$$\Rightarrow {\mathrm{x}}^6=3$$$$\Rightarrow \mathrm{x}=\sqrt[6]{3}$$

Answered by Frix last updated on 16/Jan/23

$$x>0$$$$x=t^{\frac{1}{6}}$$$$t^{\frac{t}{3}}=3\iff t\ln t=3\ln 3$$$$\mathrm{Obviously}t=3\Rightarrow$$$$x=\sqrt[6]{3}$$

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