Start with the given equation:

\(x = \ln(2y - 3) - \ln(y + 4)\)

Use the logarithm property \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\):

\(x = \ln \left( \frac{2y - 3}{y + 4} \right)\)

Exponentiate both sides to remove the logarithm:

\(e^x = \frac{2y - 3}{y + 4}\)

Cross-multiply to solve for \(y\):

\(e^x (y + 4) = 2y - 3\)

Expand and rearrange terms:

\(e^x y + 4e^x = 2y - 3\)

\(2y - e^x y = 3 + 4e^x\)

Factor out \(y\):

\(y(2 - e^x) = 3 + 4e^x\)

Solve for \(y\):

\(y = \frac{3 + 4e^x}{2 - e^x}\)