Answer
$x=\ln\frac{3}{2}$
Work Step by Step
Let $u=e^x$
$2e^{2x}+e^x-6=0$
$2u^2+u-6=0$
Factor.
$(2u-3)(u+2)=0$
Equate each factor to 0.
$u=\frac{3}{2}$ or $u=-2$
Since $e^x\gt0$ for all values of $x$
Choose only:
$u=\frac{3}{2}=e^x$
Take the natural log of both sides.
$x=\ln\frac{3}{2}$