# Using $\ln$ or $\log$

In my paper, I have $\ln$, $\log$ and $\log_2$. But, I have used both $\ln$ and $\log$ interchangeably. I wanted to be consistent and use just one of them. Which one is better?

1) I choose $\ln$, then I have to use it when stating the results in terms of Big O, e.g., $O(n \ln{n})$. Is that OK? I have seen that when people state order-results, they use $\log$ in Big O.

2) I choose $\log$, and the first time I use it, I mention that the base of the logarithm in this paper is by default $e$.

Which one is better? If I want to use the second solution, what is the best way to state that?

• I guess it depends on the field look at other papers published where you intend to publish and follow them – kjetil b halvorsen Aug 23 '17 at 19:10
• Unless the context of the paper requires a specific base, I don't think it matter which approach you use. – Wintermute Aug 23 '17 at 19:10
• If you're using $\log$ in exactly one way in your paper, it's usually best to just stick with $\log$. In the case that you switch between a natural base and another (say, base 2 in information-theoretic papers), then you can make the $\ln$/$\log$ distinction. But it's also just good to look at papers in your field and see what is the usual case. – Guillermo Angeris Aug 23 '17 at 19:11
• Use $\ln$ when you mean $\log_e$, and use $\log_a$ when the base is "a \neq e". Of course, you can explicitly define what you take $\log$ to be, but if you interchange $\log$ (predefined to be $\log_e = \ln$, then the only justified way to use $\log$ with a base other than $e$ would be to explicitly subscript the bases for distinction. I happen to think that $\ln x \cdot \log_a x$, e.g., is much more clearly disambiguated. – Namaste Aug 23 '17 at 19:13
• It's standard to use $\log$ in papers. What it says, implicitly, is "look at me, I'm a real, pure mathematician, not a dastardly engineer". – MathematicsStudent1122 Aug 23 '17 at 19:30

I suggest you use $\ln$ in the paper and $\log$ when you have order results (since the base of $\log$ does not matter, but it is conventional to use $\log$ when stating order results).
All log functions differ by a multiplicative constant. Therefore, notations like $O(n \log n)$ are unambiguous, because the $O$ absorbs multiplicative constants.
For any applicable base $b$ you have $$\log_b x = \frac 1{\ln b}\cdot\ln x$$ so $O(\log_b f(n))$ is exactly the same as $O(\ln f(n))$.
Sometimes $\log$ is used as a 'general' logarithm with the base unspecified, although constant and greater than $1$ (which is useful when multiplicative constant does not matter), while $\ln$ is log with base $e$ and $\lg$ is a logarithm with base $10$.
Sometimes $\log$ has a default base $2$, for example in computer science.
• Is that common to use $\ln$ in Big-O? It is indicating a base, however, it is not necessary. I think it is not common. – Susan_Math123 Aug 23 '17 at 19:58
• @Su20200 You're right: it is not common, because the base doesn't matter, it only introduces the constant factor which is dropped byBig-O anyway. So the symbol $\log$ is most appropriate, which denotes 'any, constant base'. – CiaPan Aug 24 '17 at 6:18