To solve the inequality \(|3^x - 8| < 0.5\), we consider the two cases:

1. \(3^x - 8 < 0.5\)

2. \(3^x - 8 > -0.5\)

For the first case:

\(3^x < 8.5\)

Taking logarithms, \(x < \log_3(8.5)\)

For the second case:

\(3^x > 7.5\)

Taking logarithms, \(x > \log_3(7.5)\)

Calculating the logarithms to 3 significant figures:

\(\log_3(8.5) \approx 1.95\)

\(\log_3(7.5) \approx 1.83\)

Thus, the solution is \(1.83 < x < 1.95\).