# What kind of logarithm does $\log$ mean in Wikipedia articles?

Sorry if this seems like a dumb question, but what what type of logarithm is $\log$ in Wikipedia articles? Cheers.

I think it depends on the situation. For a mathematician, $\log n$ probably means the natural logarithm. For a computer scientist, $\log n$ probably denotes the base $2$ logarithm, etc.

I think sometimes people write $\ln x$ for the natural logarithm, $\lg x$ for the base $2$ logarithm and $\log x$ for the base $10$ logarithm.

For programming languages such as Matlab, Python, Julia, $\log x$ means the natural logarithm of $x$.

Do you have a specific Wikipedia article that contains $\log$?

• – Robert Hanigan Aug 18 '18 at 9:37
• I think for that article, the base of the logarithm does not matter. See here. – zdm87 Aug 18 '18 at 10:07
• I deleted your "This is intended to be a comment" disclaimer, since this is the answer. – David Richerby Aug 18 '18 at 10:20
• For a computer scientist, $\log$ doesn't always mean base-$2$. Like everything else, it depends on context. For example, I'm a computer scientist but I work with probability as well as numbers of bits so I would guess I actually use natural and base-2 logs equally often. – David Richerby Aug 18 '18 at 10:23
• @zbir Thanks for a nice paper(I use Stanford Information Retrieval). That's what come in to my mind, after I posted it, that it may not matter:) – Robert Hanigan Aug 18 '18 at 15:55

It depends entirely on context. In mathematics as a whole, $\log$ usually denotes the natural logarithm (base $\mathrm{e}$). In computer science, the situation isn't as clean because we often want to talk about things like a number of bits or the height of a binary tree and, in those cases, the most natural (*baddum-tsh*) logarithm to use is base-$2$. Some authors use $\ln$ and $\lg$ to explicitly denote base-$\mathrm{e}$ and base-$2$ logs, respectively; many authors just write $\log$ to mean whatever base is most appropriate for the situation.

Of course, Wikipedia is written by a huge number of unconnected authors, so you can't assume any consistent convention is used. Indeed, a careful Wikipedia editor would make sure that $\log$ is used consistently within an article, but you can't rely on everyone being careful.

It usually doens't matter. Changing the base of a logarithm just multiplies the answer by a constant factor because, for all $a,b>1$ and all $x>0$, $$\log_a x = \frac{\log_b x}{\log_b a}\,.$$ In many situations, this multiplicative factor of $1/\log_b a$ isnt at all important. In the context of a big-$O$, big-$\Theta$, big-$\Omega$ etc bound, $O(\log n)$ means the same thing regardless of the base, since big-$O$ disregards constant factors. (But note that, e.g., $2^{\log_2 x}=x$, whereas $2^{\log_3 x}=x^{1/\log_2 3}\approx x^{1.58}$ so there are situations in which the base makes a big difference.)

Your specific example. In a comment, you say you're specifically interested in the page about tf-idf. I know nothing about document retrieval but my understanding from skim-reading that article is as follows. We're interested in ranking documents according to their relevance, and the relevance of a document is given by a score of the form $x\log y$, where $x$ and $y$ are statistics of the document. Since changing the base of the logarithm just multiplies all the scores by a constant, it doesn't change the ranking order. Therefore, it doesn't matter what base you use for the logarithm in this situation.