We can rewrite $\sqrt{27}$ as $27^{\frac{1}{2}}$.
Since $27=3^3$, we have $27^{\frac{1}{2}}=(3^3)^{\frac{1}{2}}=3^{\frac{3}{2}}$.
Setting this equal to $3^{2x}$, we get $3^{\frac{3}{2}}=3^{2x}$.
Therefore, we have $\frac{3}{2}=2x$.
Solving for $x$, we find $x=\boxed{\frac{3}{4}}$.
The answer is: \frac{3}{4}
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Solve for $x$: $3^{2x} = \sqrt{27}$. Express your answer as a common fraction.

### Answer:
