The given equation is \(\log_4 x = -3\). We know that \(\log_a b = c\) is equivalent to \(a^c = b\). Using this property, we have:
$$4^{-3} = x$$
Simplifying the right-hand side, we get:
$$\frac{1}{4^3} = x$$
Since \(4^3 = 64\), we have:
$$\frac{1}{64} = x$$
Therefore, the answer is \(\frac{1}{64}\).