Convergence in probability of $\frac{1}{n \ln (n)} \sum_{k=1}^{n} X_{k}$ to $\frac{1}{\ln (2)}$ for i.i.d RVs with $ P\left(X_{1}=2^{k}\right)=2^{-k}$

Question

Let the random RVs $X_{1}, X_{2}, X_{3}, \ldots$ be independent and identically distributed such that $\mathbb{P}\left(X_{1}=2^{k}\right)=2^{-k}$ for $k=1,2,3, \ldots$. I want to prove that $$ \frac{1}{n \ln (n)} \sum_{k=1}^{n} X_{k} \text { converges in probability to } \frac{1}{\ln (2)} $$ I tried to go directly with the definition of the convergence in probability and make use of the law of large numbers but the $\mathbb{ln}(n)$ is what got me stuck

Any suggestions