# Logarithm Notation

By Susam Pal on 05 Apr 2024

We know that the natural logarithm of a number $$x,$$ i.e., the logarithm of $$x$$ to the base $$e,$$ is sometimes denoted as $$\ln x.$$ It has other notations too. For example, many mathematics textbooks just use the notation $$\log x$$ after establishing once that this notation denotes the natural logarithm. The most descriptive notation is perhaps $$\log_e x$$ but this is most definitely an overkill. I have never seen any serious textbook use this notation.

Let us focus on $$\ln x$$ again. Is it not peculiar? What does $$\ln$$ stand for really? Logarithm natural? Sounds very unnatural.

Well, as a kid I learnt that $$\ln$$ here stands for the Latin phrase "logarithmus naturalis". It is only recently that I bothered to verify if this expansion of $$\ln x$$ that I learnt as a kid is really true. The most credible discussion of this that I could find online is this thread on Mathematics Stack Exchange: math.stackexchange.com/q/1694. The answer by Dan Velleman points us to page 277 of an 1875 book Lehrbuch der Mathematik by Anton Steinhauser. Quoting the relevant portion from the page:

Man pflegt nun, um Verwechslungen dieser beiden Systeme vorzubeugen, mit log.nat. a (gesprochen: logarithmus naturalis a) oder ln . a, oder am einfachsten mit la den natÃ¼rlichen, mit log.brigg. a (gesprochen: Logarithmus briggus a) oder log.a, oder am einfachsten mit lg. a den gemeinen Logarithmus (von a) zu bezeichnen.

Translated to English, it says:

In order to prevent confusion between these two systems, people now use log.nat. a (pronounced: logarithmus naturalis a) or ln . a, or easiest with la the natural ones, with log.brigg. a (pronounced: logarithm briggus a) or log.a, or most simply with lg. a to denote the common logarithm (of a).

So it does look like what I learnt as a kid is correct and the earliest possible reference of this the Internet is able to find for us is the 1875 book quoted above.