trend model vs AR model

I understand the differences in the structure/inputs to each model.

But in the context of what we need to know for the exam, should we just know that we would use the AR model when the trend model (even after it has been linearized via natual logs - if applicable) still contains serial correlation within its residuals?

Yes. And even AR models can have serial correlation. That’s when you consider AR2/AR3 etc until you have no more serial correlation.

There. That’s better.