# Math Help - predicate logic

1. ## predicate logic

“Exponentiation on reals has no left identity.” What does this mean exactly?

2. ## Identity element

Hello thehollow89
Originally Posted by thehollow89
“Exponentiation on reals has no left identity.” What does this mean exactly?
Exponentiation on reals takes a real number, $x$, and raises a base, $a$, say, to the power of $x$. So, using $\circ$ notation, exponentiation can be defined as the binary operation:

$a\circ x = a^x$.

A left identity is an element $i$ in the domain of
$\circ$, such that $i \circ x = x, \forall x$ in the domain.

So to say that exponentiation on reals has no left identity is to say that there is no real number $i$ for which $i^x = x, \forall x \in \mathbb{R}$.