Start with the equation \(x = 4(3^{-y})\).

Take the natural logarithm of both sides: \(\ln x = \ln(4 \cdot 3^{-y})\).

Apply the logarithm of a product: \(\ln x = \ln 4 + \ln(3^{-y})\).

Use the logarithm of a power: \(\ln(3^{-y}) = -y \ln 3\).

Substitute back: \(\ln x = \ln 4 - y \ln 3\).

Rearrange to solve for \(y\): \(y = \frac{\ln 4 - \ln x}{\ln 3}\).