On infinite divisibility of powers of a gamma variable

Consider the following three different types of probability laws on [0, ∞) 1. Lognormal distribution 2. Pareto distribution (possibly inverse, i.e, power function distribution) 3. The distribution of a power {itZ{suγ}}, {itγ}εR, of a gamma distributed random variable {itZ}. If X is an r.v. distributed according to (a), (b), or (c) above, the r.v. Y = log X will have a 1. a′) Normal distribution 2. b′) Exponential distribution (possibly negative or translated) or 3. c′) Generalized normal distribution, respectively.