A positive probability law has a density function of the general form Q(x)exp(−x1/λL(x)), where Q is subject to growth restrictions, and L is slowly varying at infinity. This law is determined by its moment sequence when λ< 2, and not determined when λ> 2. It is still determined when λ= 2 and L(x) does not tend to zero too quickly. This paper explores the consequences for the induced power and doubled laws, and for mixtures. The proofs couple the Carleman and Krein criteria with elementary comparison arguments.