Start with the equation:

\(\frac{2^x + 1}{2^x - 1} = 5\)

Multiply both sides by \(2^x - 1\):

\(2^x + 1 = 5(2^x - 1)\)

Expand the right side:

\(2^x + 1 = 5 \cdot 2^x - 5\)

Rearrange the terms:

\(2^x + 1 = 5 \cdot 2^x - 5\)

\(1 + 5 = 5 \cdot 2^x - 2^x\)

\(6 = 4 \cdot 2^x\)

Divide both sides by 4:

\(2^x = \frac{6}{4} = 1.5\)

Take the logarithm base 2 of both sides:

\(x = \log_2(1.5)\)

Calculate \(x\) using a calculator:

\(x \approx 0.585\)

Thus, the solution is \(x = 0.585\) to 3 significant figures.