By the mean value theorem, for $0<a<b$, one has$$\frac{b-a}{b}\leq\ln\frac{b}{a}\leq\frac{b-a}{a}.$$
Does this imply that, for $0<b<a$, one has$$\frac{b-a}{a}\geq\ln\frac{b}{a}\geq\frac{b-a}{b}?$$
I do not think so. I would rather think that by mean value theorem, one has$$\frac{b-a}{a}\leq\ln\frac{b}{a}\leq\frac{b-a}{b}$$
