Start with the equation \(\ln(e^{2x} + 3) = 2x + \ln 3\).

Use the property of logarithms: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\).

Rearrange the equation: \(\ln(e^{2x} + 3) - \ln 3 = 2x\).

This becomes \(\ln\left(\frac{e^{2x} + 3}{3}\right) = 2x\).

Exponentiate both sides to remove the logarithm: \(\frac{e^{2x} + 3}{3} = e^{2x}\).

Multiply through by 3: \(e^{2x} + 3 = 3e^{2x}\).

Rearrange to form a quadratic in terms of \(e^{2x}\): \(3 = 2e^{2x}\).

Divide by 2: \(e^{2x} = \frac{3}{2}\).

Take the natural logarithm of both sides: \(2x = \ln\left(\frac{3}{2}\right)\).

Solve for \(x\): \(x = \frac{1}{2} \ln\left(\frac{3}{2}\right)\).

Calculate \(x\) to 3 decimal places: \(x \approx 0.203\).