Answer
$x=-\frac{2}{3}$
Work Step by Step
We are given that $\frac{1}{4}=2^{3x}$. Both of these numbers are powers of 2, so we can rewrite the equation as $2^{-2}=2^{3x}$.
We know that $2^{-2}=\frac{1}{4}=\frac{1}{2^{2}}$.
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, $-2=3x$.
Divide both sides by 3.
$x=-\frac{2}{3}$