 Marginal and conditional slopes agree for an independent log-link random intercept

For a conditional count mean $\mathbb{E}(Y\mid x,b)=\exp (b+{\beta }_0+{\beta }^{\mathsf{T}}x)$, suppose the distribution of $b$ is independent of the covariates and has finite $\mathbb{E}{e}^b$. Integrating over $b$ gives the displayed logarithmic link function mean: every slope stays unchanged, and only the intercept shifts. This is an exact property of the log link; it does not hold in this form for the logit.