The authors establish when do the moments \(E(X^h)\) , for h in some subset C of \(\mathbb R\) , uniquely identify the distribution of any positive random variable X, that is, when is \(x^h\) a separating function. The simple necessary and sufficient condition is shown to be related with the existence of the moment generating function of the random variable \(Y=log X\) . The subset C of \(\mathbb R\) is thus the set of values of h for which the moment generating function of Y is defined. Examples of random variables characterized in this way by the set of their h-th moments are given.