# Why did I learn to write the base of the logarithm differently from the rest of the world?

It only occurred to me recently, in connection with this MO posting, that the way I write the base of the logarithm is not shared by the rest of the world. I am Dutch, and I learned at school to write the base as a superscript, $^{a}\!\log b$, rather than as a subscript, $\log_a b$. Searching the web, I found one posting that indicates they write it the same way in Indonesia, and I also notice that the Dutch Wikipedia page gives both superscript and subscript notations as alternative.

So my conjecture is that the superscript notation is peculiar for The Netherlands and its (former) colonies. Is this correct, and if it is, how and when did this difference originate? Was there some influential text book that promoted this?

• Hi Carlo, would you happen to know anything about this cartoon? Jan 31, 2018 at 16:43

It helps to remember that there was no consensus notation for logarithms well into 20th century, with $$\mathrm{l}\,x$$, $$\log x$$, and $$\mathrm{Log}\,x$$ often used by different authors and in different senses. The superscript notation was not peculiar to the Netherlands, at least not originally. According to Cajori's History of Mathematical Notations:
"Another notation was proposed by Crelle, in which the base is written above and to the left of the logarithm. This notation is sometimes encountered in more recent books. Thus, Stringham denotes a logarithm to the base $$b$$ by "$$\,^b\!\log$$"; he denotes also a natural logarithm by " $$\ln$$" and a logarithm to the complex modulul [sic!] $$k$$, by " $$\log_k$$". Stolz and Gmeiner signify by "$$\,^a\log.b$$" the "logarithm of $$b$$ to the base $$a$$". Crelle in 1831 and Martin Ohm in 1846 write the base above the "$$\,\log$$". This symbolism is found in many texts, for instance in Kimbly's Elementar-Mathematik."
Crelle founded and edited in 1826-1855 a very influential Journal für die Reine und Angewandte Mathematik, commonly nicknamed Crelle's Journal, or just Crelle. Stringham's Uniplanar Algebra (San Francisco, 1893) was apparently influential in the adoption of $$\ln$$ notation for natural logarithms, see How did the notation “$$\ln$$” for “log base e” become so pervasive? Stolz and Gmeiner promoted their notation in the monograph Theoretische Arithmetik (Leipzig, 1902), their Function Theory was also influential. But Crelle was a German, Stringham was an American, and Stolz and Gmeiner were Austrians. It is possible that a Dutch textbook author took a liking to their notation, but I would expect such a text to be in Dutch.
• What does “logarithm to the complex modulus $k$” mean? Without the context, I would interpret it the same as “logarithm to base $k$”, but that can’t be it, if they use two different notations. Jun 21 at 11:35
• @EmilJeřábek Stringham defines the base $B$ and the "modulus" $k$ geometrically, and they correspond so that $\log_k$ is the same as $^B\!\log$, see his Uniplanar Algebra, p. 80. Both $B$ and $k$ are allowed to be complex numbers. In the modern notation, this is $\log_B$, and $B^k=e$, see p. 88. Earlier (p. 41ff) he separately defined logarithms with real $B$ and $k$, and introduced $\ln$ for natural logarithms as we use it now. Jun 21 at 18:43