Some context: I've been studying Chebyshev's $\psi$ - function, which claims that $\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \log p$ where $p$ is prime and $\Lambda(n)$ is the von Mangoldt function.
In my text book, I have that $\sum_{p^k \le x, k \geq 1} \frac{\log p}{p^k} = \sum_{p \le x} \frac{\log p}{p} + \sum_{p^k \le x, k \geq 2} \frac{\log p}{p^k}$.
How and why can the summation be split in this way?
Thanks in advance.
The sum over $k$ on the left is split into the terms with $k=1$ and the terms with $k\gt1$. The terms with $k=1$ give you the first sum on the right, and the terms with $k\gt1$ (thus, $k\ge2$) give you the second sum on the right.