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Question Number 29433 by Victor31926 last updated on 08/Feb/18

$$4\left({2\mathrm{x}}^2\right)=8^{\mathrm{x}}$$

Answered by mrW2 last updated on 08/Feb/18

$$\Rightarrow 8x^2=8^x$$$$\Rightarrow {\left(2\sqrt{2}x\right)}^2=8^x$$$$\Rightarrow 2\sqrt{2}x=\pm 8^{\frac{x}{2}}$$$$\Rightarrow 2\sqrt{2}x=\pm e^{\frac{x}{2}\times \ln 8}$$$$\Rightarrow 2\sqrt{2}x\times e^{-\frac{x}{2}\times \ln 8}=\pm 1$$$$\Rightarrow 4\sqrt{2}\frac{x}{2}\times e^{-\frac{x}{2}\times \ln 8}=\pm 1$$$$\Rightarrow -\ln 8\times \frac{x}{2}\times e^{-\frac{x}{2}\times \ln 8}=\mp \frac{\ln 8}{4\sqrt{2}}$$$$\Rightarrow -\ln 8\times \frac{x}{2}=W(\mp \frac{\ln 8}{4\sqrt{2}})$$$$\Rightarrow x=-\frac{2}{\ln 8}\times W(\mp \frac{\ln 8}{4\sqrt{2}})=\{\begin{array}{l}-\frac{2}{\ln 8}\times W(-\frac{\ln 8}{4\sqrt{2}})=\{\begin{array}{l}\frac{2\times 0.9613}{\ln 8}=0.9246\\ \frac{2\times 1.0397}{\ln 8}=1\end{array}\\ -\frac{2}{\ln 8}\times W\left(\frac{\ln 8}{4\sqrt{2}}\right)=-\frac{2\times 0.2783}{\ln 8}=-0.2677\end{array}$$$$i.e.thesolutionsare:$$$$x=-0.2677,0.9246,1$$

Commented by Victor31926 last updated on 08/Feb/18

$$\mathrm{wow}...\mathrm{thanks}\mathrm{a}\mathrm{lot}.$$

Commented by Victor31926 last updated on 08/Feb/18

$$\mathrm{pls}\mathrm{explain}\mathrm{wat}\mathrm{dat}\mathrm{W}\mathrm{stands}\mathrm{for}.$$

Commented by mrW2 last updated on 08/Feb/18

$$WstandsforLambertWfunction.$$$$Itisdefinedas:$$$$W\left(a\right)\times e^{W\left(a\right)}=a$$

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