Start with the equation \(3(2^{1-x}) = 7^x\).

Take the natural logarithm of both sides: \(\ln(3(2^{1-x})) = \ln(7^x)\).

Using the logarithm properties, this becomes \(\ln 3 + (1-x)\ln 2 = x\ln 7\).

Rearrange to form a linear equation: \(\ln 3 + \ln 2 - x\ln 2 = x\ln 7\).

Combine terms: \(\ln 3 + \ln 2 = x\ln 7 + x\ln 2\).

Factor out \(x\): \(\ln 3 + \ln 2 = x(\ln 7 + \ln 2)\).

Solve for \(x\): \(x = \frac{\ln 3 + \ln 2}{\ln 7 + \ln 2}\).

Simplify: \(x = \frac{\ln 6}{\ln 14}\).