We study the number of random permutations needed to invariably generate the symmetric group, $S_n$, when the distribution of cycle counts has the strong $α$-logarithmic property. The canonical example is the Ewens sampling formula, for which the number of $k$-cycles relates to a conditioned Poisson random variable with mean $α/k$. The special case $α=1$ corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed. For strong $α$-logarithmic measures, and almost every $α$, we show that precisely $\left\lceil ( 1- α\log 2 )^{-1} \right\rceil$ permutations are needed to invariably generate $S_n$. A corollary is that for many other probability measures on $S_n$ no bounded number of permutations will invariably generate $S_n$ with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula.