Consider the two regression equations: \begin{align} Y &= \alpha+\beta X+\varepsilon \\ \log Y &= \gamma+\delta X+\eta \end{align} Is there a "simple" approximated relation between the coefficient estimated using the first equation ($\hat{\beta}$) and the one got by the second ($\hat{\delta}$) that can be used to get a sense of how approximately $\hat{\delta}$ looks like without running the regression? e.g. something like $\hat{\beta} \sim \hat{\delta}/\bar{Y}$, at least for some range of values of the variables or some hypothesis on the true DGP.
It was suggested to me by a fellow student, I cannot see why this should hold but I was not able to disprove it neither.
EDIT: I found this other related question Using results of regression on raw observation values to approximate results of regression on relative change between observations (Simple, Linear), where the asker in the similar case in which the second model is a log-log: $$ \log Y=\gamma+\delta \log X+\eta $$ proposes the formula: $$ \delta=\beta\times \frac{x_T}{\bar{y}} $$ where I guess that $T$ is the last observation (it is a time series example). There is a similar answer to the one of Ami Tavory, but not much discussion too.