Notation for Logarithms.

Question

I know that,

$\log_3 (81) = ?$

means:

What is the number to which I need to raise $3$ to obtain $81$?

The answer is, $4$.

---

If I wanted to represent this operation to look like basic mathematical operations, I could have written:

$81 \star 3 = 4$

^{Note. I use star because logarithms do not have any symbol like +, -, $*$, %, and so on.}

which means, the number $3$ is operating on $81$ to obtain $4$.

In that regard, I think, the notation $\log_3 (81)$ is misleading and very hard to remember. Every time I look at this notation, I need to do an implicit interpretation in my brain to understand it.

To make things worse, the number $3$ is termed as a $Base$.

To make it clearer, it could have been written like:

$81$ $l$ $3 = 4$

If we wanted it to look like a function, we could have also written it like:

$log(81, 3)$

---

Why did the early mathematicians choose the logarithmic notation as which is in use today rather than more clearer notation which I am talking about?

Answer 1

Because in many contexts the base is fixed. Like the early mathematicians, who used 10 as the base to such an extent that it was usually not necessary to even write it.

According to MacTutor,

Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking. ... Of course from the equation $x = a^t$, we deduce that $t = \log x$ where the $\log$ is to base $a$, but this involves a much later way of thinking. Here we are really thinking of $\log$ as a function, while early workers in logarithms thought purely of the $\log$ as a number which aided calculation.

Like I said earlier, the base was generally 10. Scientists soon realized $e$ was more useful. And for computer science today, 2 is often quite pertinent.

So if in a given context you're always going to be using only one base, it may be a reasonable shortcut to simply omit the base.

If instead you need to use two or three different bases, you might decide that you don't want to spill that much ink on them, so you write them small.

But if you write the base after the operand, there could be confusion, e.g., does $\log 81_3$ mean anything at all? Hence $\log_3 81$.

Answer 2

Because you want to think of $\log_{3}$ as a function, like $\sin$ or $\cos$. So that $\log_{3}(x)$ is the inverse function of $3^{x}$.

It doesn't really get treated so much like a binary operation for a couple of reasons: the base is restricted to be $>0$, and most commonly is almost always a positive integer or $e$. Also we tend to choose a base and stick with it, so there is not really a need for finding $\log_{b}(a)$ for lots of arbitrary choices of $b$ within a single problem.

Answer 3

This is most probably not the historical reason, but using an operator instead of a function could raise expectations about associativity and distributivity, which do not hold for logarithms.