Start with the equation:

\(\frac{3^x + 2}{3^x - 2} = 8\)

Cross-multiply to eliminate the fraction:

\(3^x + 2 = 8(3^x - 2)\)

Expand the right side:

\(3^x + 2 = 8 \times 3^x - 16\)

Rearrange to isolate terms involving \(3^x\):

\(3^x + 2 = 8 \times 3^x - 16\)

\(2 + 16 = 8 \times 3^x - 3^x\)

\(18 = 7 \times 3^x\)

Divide both sides by 7:

\(3^x = \frac{18}{7}\)

Take the logarithm of both sides:

\(x \log 3 = \log \left( \frac{18}{7} \right)\)

Solve for \(x\):

\(x = \frac{\log \left( \frac{18}{7} \right)}{\log 3}\)

Calculate \(x\) to 3 decimal places:

\(x \approx 0.860\)