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

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

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

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

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

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

Calculate \(x\) to 3 significant figures: \(x = 1.09\).