Suppose we have estimated parameters $\hat{\beta} = [\hat{\beta}_{0}, \hat{\beta}_{1}, \hat{\beta}_{2}]$ from a level-level regression: $$\hat{y} = \hat{\beta}_{0} + \hat{\beta}_{1}X_{1} + \hat{\beta}_{2}X_{2}$$ I want to use these estimates to set a prior on $\gamma_{1}$ in a Bayesian log-level regression: $$\log(y) = \gamma_{0} + \gamma_{1}X_{1} + \gamma_{2}X_{2} + e$$
Is it possible to derive an expression for $\gamma_{1}$ from the $\hat{\beta}$s?