What operators can be considered as combining two elements?

Question

New to abstract algebra and was wondering if a*b can be something like $\ a^b $ or $\ log_ab$. If not please provide an explanation as to why they aren't.

Answer 1

At it's heart, a binary operation on a set $S$ is just any function $*:S\times S\rightarrow S$. So, as long as the expressions make sense for the whole domain and the domain is closed under them, a binary operation can be whatever you want. For instance, $a^b$ defines a perfectly good operation on $\mathbb R_{>0}$. It's harder to get logarithms to work as an operation, since you don't want to put negative numbers into logarithms, but sometimes the output of a logarithm is negative.

However, usually in abstract algebra, you want some nice properties to hold of the operation. For instance, a common thing to demand is that $(x*y)*z=x*(y*z)$, which is called associativity. Exponentiation fails this property since $(x^y)^z$ is not always equal to $x^{(y^z)}$. In fact, exponentiation taken alone is a rather nasty operation, not satisfying really any identities I can think of. This makes it hard to study through abstract algebra, which generally tries to derive interesting results from identities satisfies by the operation.

Answer 2

A binary operation on a set $X$ is any function taking two arguments from $X$, with given order, and assigning an element of $X$ to them, i.e. is a function $X\times X\to X$.
For a binary operation $*$ we often use the infix notation, letting $x*y:=*(x, y) $.

For exponentiation, we can take $X:=\Bbb R^{>0}$ and then $x^y\in X$ for all $x, y\in X$, so that's a binary operation.

However, for the logarithm, it's harder - if possible at all - to find a set $X$ such that $\log$ is an operation, because of the base we must have $a>0,\ a\ne1$, that is, $X\subseteq \Bbb R^{>0}\setminus\{1\} $, but then some values $\log_xy$ might become negative: outside of $X$. To ensure $\log_xy>0$ we can restrict $x, y>1$, but then we have to ensure $\log_xy>1$, and so on..