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

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- Mar 4th 2009, 05:31 PMthehollow89predicate logic
*“Exponentiation on reals has no left identity.” What does this mean exactly?*

- Mar 6th 2009, 01:06 AMGrandadIdentity element
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