To solve the inequality \(|2^x - 8| < 5\), we consider two cases:

1. \(2^x - 8 < 5\)

2. \(2^x - 8 > -5\)

For the first case, \(2^x - 8 < 5\):

\(2^x < 13\)

Taking logarithm base 2 on both sides, we get:

\(x < \log_2 13\)

For the second case, \(2^x - 8 > -5\):

\(2^x > 3\)

Taking logarithm base 2 on both sides, we get:

\(x > \log_2 3\)

Thus, combining both inequalities, we have:

\(\log_2 3 < x < \log_2 13\)

Calculating the logarithms, we find:

\(\log_2 3 \approx 1.58\) and \(\log_2 13 \approx 3.70\)

Therefore, the solution is \(1.58 < x < 3.70\).