Given $$\lim\_{x\to \infty} \frac 1x \sum\_{n\le x} a\_n = k,$$
I want to prove that 
$$\lim\_{x\to \infty} \frac {1}{\log x} \sum\_{n\le x} \frac {a\_n}{n} = k.$$

I'm much more interested in learning the technique(s) necessary in order to prove this rather than a direct proof. Specifically, we have learned about asymptotic estimates and summation by parts, but I don't see how I can use those techniques to prove the problem statement. Thank you for any insight!