A curious identity summing over primes in an interval.

Question

Let $K \geq1$ be an arbitrary positive constant. Why do we have the equality $$\sum_{x/K \leq p \leq x} \dfrac{log(p)}{p} = log(K)+O(1),$$ where the error term is uniform in $K?$ Here the summation is over the primes in the given interval.