Simplifying an expression with e

Simplify the expression $\displaystyle e^{(\ln3) +2 \ln x}$.

My work (I feel like it's not legal!):

$\displaystyle \ \ln(e^{ln3+2 \ln x}) $

$\displaystyle = \ln3+2 \ln x $

$\displaystyle = 2\ln (3x) $

$\displaystyle = \ln (3x^2) $

$\displaystyle = e^{ln3x^2} $

$\displaystyle = 3x^2$

Am I allowed to just take the natural log of an expression when simplifying since there's no other side to take the natural log of? Same question for bumping the expression up into the exponent of e?