When do we use an autoregressive series like ln(Sales)t = b0 + b1*ln(Sales)t-1 and the series that uses time as the independent variable like ln(Sales)t = b0 + b1(t) ?

Shouldn’t they be roughly the same when graphed? As for the AR(1) model, why is the x-axis labeled with periods (t=1995, 1996, 1997…etc), and not ln(sales)t-1?

Autoregressive models are used when you assume that past values are able to predict current/future values of the dependent variable. ln(sales) t-1 = Plugging in a sales amount

When using a linear trend model ln(Sales)t = bo+b1(t), you are assuming that the dependent variable changes at a constant amount with time. b(t) = plugging in a time period

Nope, the second one is an exponential growth rate which is also constant over time (b1) and the (t) factor just count for the periods of time. Assume (b0) is zero, if not, look at it as a floor value where b1(t) starts growing.

The first one is just a transformation in order to smooth the serie to get a more linearized relation between sales T and sales T-1. Remember OLS estimation (ordinary least squares regression) has many assumptions and one of them is that there exists a linear relation between the dependent variable and the independent variable , so since sales T and sales T-1 is the same series but lagged 1 period you must LN both in order to smooth them and get a linear relation (a better one than before).