To solve the equation \(\ln(1 + 2^x) = 2\), we first remove the logarithm by exponentiating both sides:

\(1 + 2^x = e^2\).

Next, solve for \(2^x\):

\(2^x = e^2 - 1\).

Now, take the logarithm base 2 of both sides to solve for \(x\):

\(x = \log_2(e^2 - 1)\).

Using a calculator, we find:

\(x \approx 2.676\).