In reply to Ksharp.

I think it is clearer if you use Y as the target. The first model (generalized) says that each observed Y value is of the form Y = exp(X`*beta) + epsilon. So clearly the "noise" affects the response in a linear fashion. The second model (OLS of log(Y)) says that each observed Y is of the form Y = exp(X`*beta + epsilon). If you define c = exp(epsilon), then Y = c*exp(X`*beta). Thus the (transformed) noise affects the response multiplicatively.