For an AP: \(2\log_3(2^x - 5) = \log_3 2 + \log_3\left(2^x - \frac{7}{2}\right)\)

Let \(t = 2^x\): \((t - 5)^2 = 2\left(t - \frac{7}{2}\right) = 2t - 7\)

\(t^2 - 12t + 32 = 0 \Rightarrow t = 4\) or \(t = 8\)

\(t = 4\) gives \(2^x - 5 = -1 < 0\), where the log is undefined. So \(t = 8\), and \(x = 3\).

Answer: (C) 3