I have a time series dataset on UK consumption. I can (1) estimate an exponential trend (2) take the logarithm of UK consumption, estimate a linear trend for this logarithm, and exponentiate the result Why cannot I compare AIC values from these two methods?
In short, because AIC takes the log of the max value of the likelihood function. Except under strong conditions there is no a priori reason to assume that these values will be the same after log-transforming your model. In fact, a priori one should assume they will not be the same, since one of the reasons people log-transform their models is to change the sample distribution to be a little closer to normal.
So, you can't simply look at the two AIC values and conclude that they convey the same information, because in general, they do not.