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?
Why cannot I compare AIC values obtained from nonlinear least squares and the ordinary least squares?
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.