Bounding a difference by the logarithm of a fraction

Question

Can it be shown that there exists a constant $c(M)$ such that $$x - y \leq c(M) \cdot \log\left(\frac{x}{y}\right)$$ for all $x, y \in (0,M]$ and $x \geq y$? It appears to hold for $c(M) = 2.31 \cdot M$ experimentally (https://www.desmos.com/calculator/j6zgcngmll).

Answer 1

Your observation follows from the Mean Value Theorem, which says there is a $z$ between $y$ and $x$ such that

$$\frac{1}{\ln(10)}\cdot\frac{1}{z} = \frac{\log_{10}(x)-\log_{10}(y)}{x-y} = \frac{\log_{10}(x/y)}{x-y}.$$