Solve ${2}^{x}={x}^{2}$

I've been asked to solve this and I've tried a few things but I have trouble eliminating $x$. I first tried taking the natural log:

$x\mathrm{ln}\left(2\right)=2\mathrm{ln}\left(x\right)$

$\frac{\mathrm{ln}\left(2\right)}{2}}={\displaystyle \frac{\mathrm{ln}\left(x\right)}{x}$

I don't know what to do from here so I decided to try another method:

${2}^{x}={2}^{{\mathrm{log}}_{2}\left({x}^{2}\right)}$

$x={\mathrm{log}}_{2}\left({x}^{2}\right)$

And then I get stuck here, I'm all out of ideas. My guess is I've overlooked something simple…

I've been asked to solve this and I've tried a few things but I have trouble eliminating $x$. I first tried taking the natural log:

$x\mathrm{ln}\left(2\right)=2\mathrm{ln}\left(x\right)$

$\frac{\mathrm{ln}\left(2\right)}{2}}={\displaystyle \frac{\mathrm{ln}\left(x\right)}{x}$

I don't know what to do from here so I decided to try another method:

${2}^{x}={2}^{{\mathrm{log}}_{2}\left({x}^{2}\right)}$

$x={\mathrm{log}}_{2}\left({x}^{2}\right)$

And then I get stuck here, I'm all out of ideas. My guess is I've overlooked something simple…