Lets say I have a random variable $y$ that is expected to be:
$$y_i \sim poisson(\lambda_i)$$
$$log(\lambda_i) = \beta_0 + \beta_1x_i$$
But, for every reasons, I am running the following linear regression using OLS:
$$log(y_i) = \beta_0 + \beta_1log(x_i)$$
Given the log-log transformation, I could than say "a 1% change in $x$ causes a $\beta_1$% change in $y$". Switching back to the poisson context, how would I set up a regression model where a similar interpretation (% change in $x$ causes $\beta_1$% change in $y$) is possible?
Since the log is the common link-function for a poisson glm, I do not need to transform the lhs of the equation. So transforming $x_1$ should be sufficient?