Start with the equation \(\ln(1 + e^{2y}) = x\).

Exponentiate both sides to remove the natural logarithm: \(1 + e^{2y} = e^x\).

Rearrange to isolate \(e^{2y}\): \(e^{2y} = e^x - 1\).

Take the natural logarithm of both sides: \(2y = \ln(e^x - 1)\).

Divide by 2 to solve for \(y\): \(y = \frac{1}{2} \ln(e^x - 1)\).