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