Is $x^x$ an exponential function?

Question

I know that functions of the form $c^x$ are called exponential when $c$ is a constant.

How about the function $x^x$? It seems somewhere in between exponential and double exponential to me. Is there a good way to describe it?

Answer 1

Regarding "describing the function": note that $$ x^x = e^{x \ln x} $$ So indeed, we may state that (asymptotically) $e^x\leq x^x \leq e^{x^2}$.

Answer 2

It is not an exponential function. A function is called exponential if $f(x) = a^x$ where $a > 0$ ,$a \ne 1$, and $a$ is constant.

Answer 3

For the fundamental operations: addition, subtraction, multiplication, division, and exponentiation one may ask what result is achieved if one applies one of those operation to a number itself. To add a number to itself is called doubling, to subtract a number from itself is nameless since it yields always zero, to multiply a number by itself is called squaring, to divide a number by itself remains nameless since it yields one (except $0/0$). For doubling and squaring there are numerous practical reasons.

A number powered to itself is practically useless, one may call it auto-exponentiation or a self-power, but nobody cares.

Edit (to explain the naming): one may consider $x\cdot x$ as a product, it's also $x^2$, that is, a square.