Is it possible to solve for $x$ using the lambert W function in the expression ${\ln\left(x\right)}=(t-x)^2$?

Question

${\ln\left(x\right)}=(t-x)^2$

$\pm\sqrt{\ln\left(x\right)}+x=t$

$\mathrm{e}^{\sqrt{\ln\left(x\right)}+x}=e^t$

And that is as close as I can get it to the form $x\mathrm{e}^x$. What do I do next? Is it possible to solve it this way ? Are there any generalizations?