I have in my problem that $X_1,\ldots,X_n$ is a random sample from a distribution with probability density $f(x; \theta)=\theta x^{\theta-1}, 0<x<1$. Furthermore, $-\log X_i \sim\text{EXP}(1/\theta), \,\, i=1,\ldots,n$.
One of the intermediary steps in my problem just states: "By the WLLN, $$-\frac{1}{n}\sum_{i=1}^n \log X_i \xrightarrow{p}E[-\log X_i]=\frac{1}{\theta}"$$
I feel like this using some other limit theorem as well with the WLLN. Perhaps someone could guide me as to which one is being used?