Sometimes derivatives are (inappropriately) interpreted and handled as if they were fractions. For instance, when applying a chain rule, $df/dx$ would be handled as if $df$ was the numerator and $dx$ the denominator. Often, though, and especially in simple cases, the results turn out to be the correct ones.
In the same spirit of those uncomfortable practises, is there anything looking like $$\log\frac{df}{dx}=\log df - \log dx$$ that could too (sometimes, somehow) make sense?