Link functions in poisson regression I've recently started studying statistics and a question came up to my mind while reading about poisson regression:
If we have to exponentiate all terms in order to have only positive values, why do we use a log function as a link?
I'm sorry if this seems a silly question, i hope someone can help me understand this topic better.
Thanks in advance

$$
\log\bigg(\mathbb E\big[Y\vert X\big]\bigg) = X\beta\\
\Big\Updownarrow\\
\mathbb E\big[Y\vert X\big] = \exp\big(X\beta\big)
$$
The link function is what is applied to the expected response, not what is applied to the linear part of the regression equation to calculate the expected response. In general for a generalized linear model:
$$
\text{link}\bigg(\mathbb E\big[Y\vert X\big]\bigg) = X\beta
$$
It seems that you understand the concepts but just missed this bit of terminology.
