Start with the equation \(4|5^x - 1| = 5^x\).

This implies two cases: \(4(5^x - 1) = 5^x\) and \(4(1 - 5^x) = 5^x\).

For the first case, \(4(5^x - 1) = 5^x\):

\(4 \times 5^x - 4 = 5^x\)

\(3 \times 5^x = 4\)

\(5^x = \frac{4}{3}\)

Taking logarithms, \(x = \log_5\left(\frac{4}{3}\right)\).

For the second case, \(4(1 - 5^x) = 5^x\):

\(4 - 4 \times 5^x = 5^x\)

\(4 = 5^x + 4 \times 5^x\)

\(4 = 5 \times 5^x\)

\(5^x = \frac{4}{5}\)

Taking logarithms, \(x = \log_5\left(\frac{4}{5}\right)\).

Calculating these values gives \(x = 0.179\) and \(x = -0.139\).