Notation for Logarithms.

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.

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 $\log_3 81$.