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

\(\ln(5^{3-2x}) = \ln(4 \cdot 7^x)\)

Apply the logarithm of a product rule: \(\ln(a \cdot b) = \ln a + \ln b\):

\(\ln(5^{3-2x}) = \ln 4 + \ln(7^x)\)

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

\((3-2x) \ln 5 = \ln 4 + x \ln 7\)

Expand and rearrange to form a linear equation in \(x\):

\(3 \ln 5 - 2x \ln 5 = \ln 4 + x \ln 7\)

Combine like terms:

\(3 \ln 5 - \ln 4 = x \ln 7 + 2x \ln 5\)

Factor out \(x\):

\(3 \ln 5 - \ln 4 = x(\ln 7 + 2 \ln 5)\)

Solve for \(x\):

\(x = \frac{3 \ln 5 - \ln 4}{\ln 7 + 2 \ln 5}\)

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

\(x \approx 0.666\)