Can $\frac{y - x}{\ln y-\ln x}$ be rational, for positive reals $x$ and $y$?

Question

Consider two positive real numbers $x$ and $y$ such that $y > x \geq 0$. Do there exist pairs $\left\{x,y\right\}$ such that

$$ \frac{y - x}{\ln\left(\frac{y}{x}\right)} = \frac{y-x}{\ln(y)-\ln(x)} \in \mathbb{Q}_+, $$

where $\mathbb{Q}_+$ is the set of positive rational numbers?

Answer 1

The map

$$f_x(y) = \frac{y-x}{\ln(y)-\ln(x)}$$ defined on $(x, \infty)$ is continuous and positive. As it is not constant, it takes rational values according to the intermediate value theorem as $\mathbb Q$ is dense in $\mathbb R$.