Answer
$x=-\frac{3}{2}$
Work Step by Step
We are given that $\frac{1}{27}=3^{2x}$. Both of these numbers are powers of 3, so we can rewrite the equation as $3^{-3}=3^{2x}$.
We know that $3^{-3}=\frac{1}{27}=\frac{1}{3^{3}}$.
From the uniqueness of $b^{x}$, we know that $b^{x}=b^{y}$ is equivalent to $x=y$ (when $b\gt0$ and $b\ne1$).
Therefore, $-3=2x$.
Divide both sides by 2.
$x=-\frac{3}{2}$
