SCHWEITZER 1977 Problem 3

Prove that if $a,x,y$ are $p$ -adic integers different from $0$ and $p | x, pa | xy$ , then $$\frac 1y \frac{(1+x)^y-1}{x} \equiv \frac{\log (1+x)}{x} \;\;\; ( \textrm{mod} a\ ) .$$

L. Redei