$\log(x)$ or $\ln(x)$ to denote the natural logarithm in research papers? Since the natural logarithm, i.e. with base $e$, is very commonly used in research papers and that both $\ln(x)$ and $\log(x)$ are used to denote it, it is natural* to ask which of these notations to use when preparing a paper. The fact that both are used in literature concerning the same topics  gives rise to unnecessary confusions and/or definitions. Both have their advantages and disadvantages:


*

*The notation $\ln(x)$ bears no ambiguity, as its name is the abbreviation of the French logarithme naturel, or natural logarithm. One does not need to define what it denotes, it is self-explanatory. However, not everyone likes to use it, because...

*The notation $\log(x)$ is used much more widely for historical  reasons as well as notational conventions. However, literally every time it is mentioned in a paper, it is followed by something along the lines of "where $\log(x)$ denotes the natural logarithm, whose base value is $e$", which is not only cumbersome for the reader (who has read this phrase a hundred times before), but can also be avoided by simply using $\ln(x)$.


So which notation is best suited for denoting $\log_e(x)$ and why?

*pun not intended
 A: In number theory, the notation  $ \log $  is commonly used, especially when asymptotics are considered. One also frequently uses the notation  $ \log_{k} $ for the  $ k $ -th iterate of this function. Indeed the natural logarithm is essentially the only one that matters. This may not be true for other subfields of mathematics.
A: I would divide it into two cases:


*

*If you care about the base being $e$, use $ln$ or state that $log$ is supposed to have base $e$.

*If you only want a logarithm but don't care about which one, use $log$ (and maybe state that you don't care about the base). This case often arises when talking about growth of functions, running times of algorithms, etc, as logs with respect to different bases only differ by a constant.


In general, there is no "best solution" for your problem, otherwise there wouldn't be different versions running around. Make sure that your paper gets the message across, is correct and nice to read. If you do everything else right, using $ln$ or $log$ is then simply a question of style.
