I believe that the problem with the original question here is not that the data are not a good fit to the model, nor is it that the log needs to be taken to make the data approximate a straight line. The problem is that the fit-model is mathematically ill-defined as follows. The fit-function used is:

A*Exp[B*x + D]

Now, that exponent can always be re-written, using the elementary properties of the exponential function:

A*Exp[D]*Exp[B*x]

Then, the constant magnitude multiplying the exponential function can be seen to be the product of two constants, A and Exp[D]. One is redundant since, e.g. the latter could be absorbed into the former without loss of generality. As a consequence, the fitter cannot find a unique minimum since any variation in one is compensated by a variation of the other, and there is no unique fit. To cure, this, simply remove the "+D" from the exponential function, and your fit will run fine. The quality of the fit would remain an open question for further study.