Generally I like to write the equation as:

log(E[Count]) = \beta_0 + \beta_1(X) + 1*Offset

Sometimes people are confused when writing log(Count), as they think they need to take the logarithms of the count before the regression equation. And then different software sometimes uses the term "exposure", which is typically the raw population, whereas "offset" it should be the logged population. https://apwheele.github.io/MathPosts/PoissonReg.html

I still have the idea in the queue from an older post of yours Rick to show the difference in error distributions for Poisson/Negative Binomial/OLS of logged counts.