We start with the equation \(2|3^x - 1| = 3^x\).

This can be split into two cases:

Case 1: \(2(3^x - 1) = 3^x\)

\(2 \cdot 3^x - 2 = 3^x\)

\(2 \cdot 3^x - 3^x = 2\)

\(3^x = 2\)

Taking logarithms, \(x = \log_3 2\).

Calculating, \(x \approx 0.631\).

Case 2: \(2(1 - 3^x) = 3^x\)

\(2 - 2 \cdot 3^x = 3^x\)

\(2 = 3^x + 2 \cdot 3^x\)

\(2 = 3 \cdot 3^x\)

\(3^x = \frac{2}{3}\)

Taking logarithms, \(x = \log_3 \frac{2}{3}\).

Calculating, \(x \approx -0.369\).

Thus, the solutions are \(x \approx 0.631\) and \(x \approx -0.369\).