What is the history of the different logarithm notations?

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?

• I'm from Singapore (geographically close and not that different culturally) and I've never seen the former notation. May 24 '20 at 10:03
• This question belongs on hsm.se.
– J.G.
May 24 '20 at 10:05
• @Deepak That's why I'm asking this. I don't get why we need to use a different notation. Especially if it's not commonly used internationally. May 24 '20 at 10:07

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 $$\log_{10} x$$.")
• I don't think that your statement that "In America, $\log a$ represents base $10$" is entirely accurate. This is not a national thing, but rather an indication of what field one works in. As an American mathematician, I use $\log$ to denote the natural logarithm---this seems to be common among mathematicians. Engineers typically use $\log$ to mean the "common logarithm" (i.e. the log base $10$), and some computer scientists use $\log$ to denote the logarithm base $2$ (other use $\lg$ for this). May 24 '20 at 14:19