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Question Number 54473 by Tawa1 last updated on 04/Feb/19

$$\mathrm{Solve}\mathrm{for}\mathrm{x}:2^{2\mathrm{x}-4}={\mathrm{x}}^2$$

Answered by mr W last updated on 04/Feb/19

$${\left(2^{x-2}\right)}^2=x^2$$$$2^{x-2}=\pm x$$$$\frac{1}{4}{2}^x=\pm x$$$$\frac{1}{4}{e}^{x\ln 2}=\pm x$$$$\pm \frac{1}{4}={xe}^{-x\ln 2}$$$$\pm \frac{\ln 2}{4}=(-x\ln 2)e^{-x\ln 2}$$$$-x\ln 2=\mathbb{W}(\pm \frac{\ln 2}{4})$$$$\Rightarrow x=-\frac{1}{\ln 2}\mathbb{W}(\pm \frac{\ln 2}{4})=\{\begin{array}{l}-0.2153\\ 0.3099\\ 4\end{array}$$

Commented by Tawa1 last updated on 04/Feb/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$$$$$

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