Start with the equation \(2^{3x-1} = 5 \cdot 3^{-x}\).

Take the natural logarithm of both sides: \(\ln(2^{3x-1}) = \ln(5 \cdot 3^{-x})\).

Apply the logarithm laws: \((3x-1)\ln 2 = \ln 5 + (-x)\ln 3\).

Rearrange to form a linear equation: \(3x \ln 2 - x \ln 3 = \ln 5 + \ln 2\).

Factor out \(x\): \(x(3 \ln 2 - \ln 3) = \ln 5 + \ln 2\).

Solve for \(x\): \(x = \frac{\ln 5 + \ln 2}{3 \ln 2 - \ln 3}\).

Simplify the expression: \(x = \frac{\ln 10}{\ln 24}\).