This may be more of a pun than anything else, as the two usages of the word "natural" are unrelated, but I was wondering if there was any nontrivial sense in which the natural logarithm could be viewed as an example of naturality.
Many things seem to be reframeable in the context of category theory. In the category of presheaves on $\mathbf{Set}$, the natural logarithm is indeed a natural transformation under the Yoneda embedding, but so is every other set function, so I would not consider this example to showcase a particular naturality of the natural logarithm. A sufficient construction would likely need to relate to the calculus-based properties of the natural logarithm and $e$. Ideally, does there exist a nice categorial context in which the natural logarithm exhibits naturality but no logarithms of other bases exhibit naturality?