Interpretation difference between log link and log transformation

Question

I have a question about the interpretation difference between log link of GLM and log transformation of LM. I know that log transformation is for target variable but log link is for mean .But related with the interpretation of the model seems they are similar ? both need e^coefficent to measure the influence of the target variable? The question may be convert to if we want to get 1) estimation of Y or 2)the interpretation of coefficient, how these two affect the result ?

I have some clue that:

1. For the interpretation of coefficient : using mean to measure the impact ? so for the log of LM, nothing changing with normal distribution , so u=p0+p1x. For log link will be u=e^(po+p1x)
2. For the estimation of y, using signal function and estimator of p to get y.For log transform y=e^(po+p1x), for log link is only depend on the which exponential family choose.

But I am not very clear why using mean to moniter the impact of coefficient. I originally thought is same as the estimation of y. So I am confused there

Answer 1

Those models are similar, but the key different thing is we model $\log{E(Y)}$ for GLM and $E(\log Y)$ for LM. Thus, we can estimate $Y$ directly in GLM and $\log Y$ in LM.

GLM: We can directly say $E(Y)=\exp(\beta_0+\beta_1X)$. In this case, $\beta_1$ catches the effect of a unite change in $X$ on $Y$ (acutually, log of ratio of $Y$).

LM: we only say $E(\log Y)=\beta_0+\beta_1X$. Then, $\log Y$ increases by $\beta_1$ when $X$ increases by 1.

Critical note: $E(\log Y)\neq \log E(Y) $.