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