The equation given is \(|4 - 2^x| = 10\). This implies two cases:

1. \(4 - 2^x = 10\)

2. \(4 - 2^x = -10\)

For the first case:

\(4 - 2^x = 10\)

\(-2^x = 10 - 4\)

\(-2^x = 6\)

This case is not possible as \(2^x\) is always positive.

For the second case:

\(4 - 2^x = -10\)

\(-2^x = -10 - 4\)

\(-2^x = -14\)

\(2^x = 14\)

Taking logarithms on both sides:

\(x \log 2 = \log 14\)

\(x = \frac{\log 14}{\log 2}\)

Calculating the value:

\(x \approx 3.81\)