Particular inequality equivalence

Question

Why is $\sum_i^M ln(r_i)(r_i^{'}-r_i) \leq 0$ equivalent to $\sum_i^M \frac{r_i^{'}-r_i}{r_i} \leq 0$ for a finite $M>0$ and positive, real-valued $r_i, r_i^{'}$?

Answer 1

Let $r_i=\frac{1}{e}>0$ and $r'_i=2>0$.

Then, as $\log \frac{1}{e} = -1$, the first sum would always be negative and equal $-M(2-\frac{1}{e})<0$.

On the other hand, the second sum would be

$$\sum^M \frac{2-\frac{1}{e}}{\frac{1}{e}}=Me \left ( 2-\frac{1}{e} \right ) >0 $$