Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$

${10}^{-3}{x}^{{\mathrm{log}}_{10}x}+x({\mathrm{log}}_{10}^{2}x-2{\mathrm{log}}_{10}x)={x}^{2}+3x$

To be fair I wasn't able to make any progress. I tried using substitution for the logarithms, but it didn't help at all.

This is a contest problem, so there should be a nice solution.

Any help? Clue?

Solve the equation in $\mathbb{R}$

${10}^{-3}{x}^{{\mathrm{log}}_{10}x}+x({\mathrm{log}}_{10}^{2}x-2{\mathrm{log}}_{10}x)={x}^{2}+3x$

To be fair I wasn't able to make any progress. I tried using substitution for the logarithms, but it didn't help at all.

This is a contest problem, so there should be a nice solution.

Any help? Clue?