Home -> Solved problems -> Can you solve it? x^2 = 2^x


\[\begin{aligned} &x^{2}=2^{x} \\\\ &x\neq0\\\\ &\ln {\left(x^{2}\right)} = \ln \left(2^{x}\right)\\\\ &2 \ln |x|=x \ln (2)\\\\ &x^{-1} \ln |x|=\frac{1}{2} \ln (2)\\\\ &\text { case (1) :}\;\;\;\;\;\;\; x>0\\\\ &x^{-1} \ln (x)=\frac{1}{2} \ln (2)\\\\ &\text { Using Lambert W function }\\\\ &W\left(-\ln (x) e^{-\ln x}\right)=W(-\ln (\sqrt{2}))\\\\ &-\ln (x)=W(-\ln \sqrt{2})\\\\ &x=e^{-W(-\ln \sqrt{2})} \end{aligned}\]
\[\begin{aligned} &\text { case (2) :}\;\;\;\;\;\;\; x<0\\\\ &(-x)^{-1} \ln (-x) =-\frac{1}{2} \ln (2) \\\\ &W\left(-\ln (-x) e^{-\ln (-x)}\right)=W(\ln \sqrt{2}) \\\\ &-\ln (-x) =W(\ln \sqrt{2}) \\\\ &x =-e^{-W(\ln \sqrt{2})} \end{aligned} \]
Home -> Solved problems -> Can you solve it? x^2 = 2^x

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