Answer
$x=\displaystyle \frac{3}{2}$
Work Step by Step
$3^{2x-1}=3^{2x}\displaystyle \cdot 3^{-1}=\frac{1}{3}\cdot 3^{2x}$, so the equation is
$ 3^{2x}-\displaystyle \frac{1}{3}\cdot 3^{2x}=18\qquad$ ... factor out $3^{2x}$
$3^{2x}(1-\displaystyle \frac{1}{3})=18$
$ 3^{2x}(\displaystyle \frac{2}{3})=18\qquad$ ... multiply with $\displaystyle \frac{3}{2}$
$3^{2x}=\displaystyle \frac{18\cdot 3}{2}$
$3^{2x}=27$
$ 3^{2x}=3^{3}\qquad$ ... apply the exponential equality principle
$2x=3$
$x=\displaystyle \frac{3}{2}$
