Is $x^x$ an exponential function? 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?
 A: Regarding "describing the function": note that
$$
x^x = e^{x \ln x}
$$
So indeed, we may state that (asymptotically) $e^x\leq x^x \leq e^{x^2}$.
A: 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.
A: 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.
A: I'm not too sure about this, but I think it would be a 2nd tetration of x. Whether this works like an exponential function, I'm not certain about, but it is commonly referred to as "iterated exponentiation", and looks to have similar properties. 
Here's the Wikipedia link: https://en.wikipedia.org/wiki/Tetration
