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

\(\ln(4^{3x-1}) = \ln(3 \cdot 5^x)\)

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

\((3x-1) \ln 4 = \ln 3 + x \ln 5\)

Expand and rearrange the equation to solve for \(x\):

\(3x \ln 4 - \ln 4 = \ln 3 + x \ln 5\)

\(3x \ln 4 - x \ln 5 = \ln 3 + \ln 4\)

Factor out \(x\):

\(x(3 \ln 4 - \ln 5) = \ln 3 + \ln 4\)

Solve for \(x\):

\(x = \frac{\ln 3 + \ln 4}{3 \ln 4 - \ln 5}\)

Calculate the value of \(x\) to three decimal places:

\(x \approx 0.975\)