you could use a numerical approach like newton's method to find a solution. if I am not mistaken you can say that x>0 since ln(x) is undefined otherwise and then rewrite your equation as [itex]x^{3}e^{2x}-e^{6}=0[/itex]
an approximate solution is possible through pt of intersection of the 2 graphs...but i dont think theres any way to find an exact solution save hit n trial