Start by taking the natural logarithm of both sides of the equation:

\(\ln(2^{3x-1}) = \ln(5 \cdot 3^{1-x})\)

Apply the logarithm power rule: \(\ln(a^b) = b \ln a\).

\((3x-1)\ln 2 = \ln 5 + (1-x)\ln 3\)

Expand and rearrange the equation:

\(3x \ln 2 - \ln 2 = \ln 5 + \ln 3 - x \ln 3\)

Combine like terms:

\(3x \ln 2 + x \ln 3 = \ln 5 + \ln 3 + \ln 2\)

Factor out \(x\):

\(x(3 \ln 2 + \ln 3) = \ln 5 + \ln 3 + \ln 2\)

Solve for \(x\):

\(x = \frac{\ln 5 + \ln 3 + \ln 2}{3 \ln 2 + \ln 3}\)

Combine the logarithms:

\(x = \frac{\ln(5 \cdot 3 \cdot 2)}{\ln(2^3 \cdot 3)}\)

\(x = \frac{\ln 30}{\ln 24}\)