“Exponentiation on reals has no left identity.” What does this mean exactly?
Hello thehollow89Exponentiation on reals takes a real number, $\displaystyle x$, and raises a base, $\displaystyle a$, say, to the power of $\displaystyle x$. So, using $\displaystyle \circ$ notation, exponentiation can be defined as the binary operation:
$\displaystyle a\circ x = a^x$.
A left identity is an element $\displaystyle i$ in the domain of $\displaystyle \circ$, such that $\displaystyle i \circ x = x, \forall x$ in the domain.
So to say that exponentiation on reals has no left identity is to say that there is no real number $\displaystyle i$ for which $\displaystyle i^x = x, \forall x \in \mathbb{R}$.
Grandad