Completely asymmetric stable laws and Benford's law

Let $Y$ be a random variable with a completely asymmetric stable law and parameter $\alpha$. This paper proves that a probability distribution of a fractional part of the logarithm of $Y$ with respect to any base larger than 1 converges to the uniform distribution on the interval $[0,1]$ for $\alpha\to 0$. This implies that the distribution of the first significant digit of $Y$ for small $\alpha$ can be approximately described by the Benford law.