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

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-Archimedes-

### The Beauty of Mathematics : It Can Never Lie to You

Mathematics is the queen of the sciences.

#### Carl Friedrich Gauss

To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.

#### Leonhard Euler

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

#### Pythagoras

Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.