What is the history of the different logarithm notations?

Question

There are two different notation of writing logarithms. In my country (Indonesia), the notation of writing logarithms is

$$ ^a\log b $$

but the most commonly used notation is

$$ \log_{a}b $$

I'm used to the widely used notation. I've searched the web for why there are two different notation, but I couldn't find any. This is only thing I could find relating to that which is what I've explained above that I already know. Are there any historical reasons for this?

Answer 1

Based on the comment by @J.G., I researched to the HSM Stack Exchange website. I found this answer there. It is the same book that is also suggested by @bjcolby15. Apparently, it is also widely used in the Netherlands. Considering that Indonesia was colonized by the Dutch, I think the most reasonable explanation is the Indonesians adapted the notation from the Dutch.

Answer 2

Try this - I found it on the Internet Archive...History of Mathematical Notation, Volume II...the link will bring you right to the section related to logarithms.

I like the Indonesian version $^a\log b$ because it indicates the base a lot more clearly (versus the current convention $\log_a b$), but when they started typesetting mathematical books in the $16$th century, the writers discovered that the base might have been misunderstood as a power (i.e. $b^a$), so to avoid confusion, the current convention of placing the base under the logarithm was used.

Another convention: some mathematicians will use $\log a$ for different bases. For instance, $\log a$ could represent the natural logarithm $\ln a$, the common logarithm $\log_{10} a$, or any base they choose. For the layman, it's confusing unless the base is specifically defined (i.e. "Throughout this book, $\log x$ means the natural logarithm, often denoted $\ln x$; the common logarithm will be represented by $\log_{10} x$.")