# Can a GLM with exponential response distribution be transformed into a Poisson regression instead?

Given the following generative model

$$T | \mathbf{x_i} \sim \operatorname{Exp}(\lambda_i)$$ $$\ln(\lambda_i) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots$$

That is: I have observations of covariate/response pairs $(\mathbf{x}_i, t_i)$, such that $t_i$ are measurements of inter-arrival time of events, and is assumed to be exponentially distributed. I would like to fit a GLM such that the response distribution is exponential and using a log link. I note that you can do it in a roundabout way using a gamma family: Fitting exponential (regression) model by MLE?

What I am wondering is whether it is possible to transform this into a Poisson regression model like the following:

$$Y | \mathbf{x_i} \sim \operatorname{Pois}(\lambda_i)$$ $$\ln(\lambda_i) = \beta_0 + \beta_1 x_1 + \dots + \beta_n x_n + \ln(t_i)$$

that is, include the time to observe one event as an offset term in the regression equation, which factors in the exposure time: Where does the offset go in Poisson/negative binomial regression?