Exponential smoothing in state space form (ETS) is a popular forecasting technique, widely used in research and practice. While the additive error ETS models have been well studied, the multiplicative error ones have received much less attention in forecasting literature. Still, these models can be useful in cases, when one deals with positive data, because they are supposed to work in such situations. Unfortunately, the classical assumption of normality for the error term might break this property and lead to non-positive forecasts on positive data. In order to address this issue we propose using Log-Normal, Gamma and Inverse Gaussian distributions, which are defined for positive values only. We demonstrate what happens with ETS(M,*,*) models in this case, discuss conditional moments of ETS with these distribution and show that they are more natural for the models than the Normal one. We conduct the simulation experiments in order to study the bias introduced by point forecasts in these models and then compare the models with different distributions. We finish the paper with an example of application, showing how pure multiplicative ETS with a positive distribution works.