Take the natural logarithm of both sides of the equation:

\(\ln(2^{5x}) = \ln(3^{2x+1})\)

Apply the power rule for logarithms: \(a \ln(b) = \ln(b^a)\).

\(5x \ln(2) = (2x+1) \ln(3)\)

Expand the right side:

\(5x \ln(2) = 2x \ln(3) + \ln(3)\)

Rearrange to solve for \(x\):

\(5x \ln(2) - 2x \ln(3) = \ln(3)\)

Factor out \(x\):

\(x (5 \ln(2) - 2 \ln(3)) = \ln(3)\)

Solve for \(x\):

\(x = \frac{\ln(3)}{5 \ln(2) - 2 \ln(3)}\)

Calculate \(x\) to 3 significant figures:

\(x \approx 0.866\)