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

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

Using the logarithm power rule, this becomes \(x \ln e = (x-2) \ln 3\).

Since \(\ln e = 1\), the equation simplifies to \(x = (x-2) \ln 3\).

Rearrange to solve for \(x\):

\(x = x \ln 3 - 2 \ln 3\)

\(x - x \ln 3 = -2 \ln 3\)

\(x(1 - \ln 3) = -2 \ln 3\)

\(x = \frac{-2 \ln 3}{1 - \ln 3}\)

Calculate \(x\) using a calculator to get \(x \approx 22.281\).