Start with the given equation:

\(z = \ln(y+2) - \ln(y+1)\)

Use the law of logarithms for subtraction: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\).

Thus, \(z = \ln\left(\frac{y+2}{y+1}\right)\).

Exponentiate both sides to remove the logarithm:

\(e^z = \frac{y+2}{y+1}\)

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

\(e^z(y+1) = y+2\)

\(e^z y + e^z = y + 2\)

Rearrange terms to isolate \(y\):

\(e^z y - y = 2 - e^z\)

\(y(e^z - 1) = 2 - e^z\)

\(y = \frac{2 - e^z}{e^z - 1}\)