Why Log and not Ln ?

The common mathematical notation for the natural log is log. One writes log₁₀ for the base 10 log or log₂ for the base 2 log. Of course, you are free to use ln, but there are reasons to grow up and make the switch:

• almost all more advanced mathematics uses log. With a little bit polemic, one can say that ln is used by kids while log is used by grown ups.
• almost all programming languages use log and not ln.
• ln is a very lonely function, it has a inferiority complex as it has only two letters. Its friends sin, cos, exp, cot, tan, sec, csc all have three letters. Let it become a full member of the three letter family!
• the three letters add more clarity, especially in handwritten text. The l can be read as 1 for example. Without context ln(x) could be mixed up with 1 n(x) for example.
• Especially when programming, the letter l and the 1 are easy to mix up. The three letter notation makes things much more robust. If you write 1n[x] in Mathematica for example, this is just n[x]. Log[x] can not be confused so easily with anything else. If you have done some programming, you know that a silly thing like that can ruin your afternoon or entire night when debugging a large program.
• It is nicer to write rules like log(exp(x)) = x and exp(log(x)) = x. It is a bit a matter of taste to have symmetry between the two functions log and exp as inverse functions.
• ln(x) is just awful to spell. It sounds as if you throw up. Just spell it out and you hear it. You literally have to torture your tongue to spell it as there are no vocals.
• If you write a paper and a referee sees you use ln, it might expose you as a math newbee. Of course, judging by notation is stupid, but people do judge for stupid reasons like cloths, titles, languages. And using ln makes a statement that you have not seen much math beyond calculus or not programmed much beyond Excel which by the way has a terrible math reputation (Examples: 1 2 3 4 (and also breaks basic data coding paradigms like mixing code with data. Splitting data from code has the advantage that it can be audited better (it is almost impossible to audit an excel spread sheet) and that one can test and process the data with different programming languages).

Why is log used in serious programming languages? Because earlier in computer science, writers of programming languages were also fluent in mathematics (there was a time when math and comp sci were close together). The language designers knew what they were doing and were not obsessed with the 10 fingers on their hands. There is of course also some historical reason. Fortran for example used LOG and many programming languages followed.

Programming languages

• Mathematica: Log
• Matlab: log
• Java: Math.log
• Javascript: Math.log
• C: log
• Python: math.log
• Perl: log
• Pari: log
• R: log
• Excel: uses LOG as base 10 log

Higher math

Here are a few screen shots of books with a bit more advanced math than calculus. Examples of books where ln is used are basic calculus books Stewart.

Ahlfors

Apostol

Conway

Landau

Lang

Loomis Sternberg

Rudin

Edwards

Stopple

Feynman