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?

You can describe its rate of growth as superexponential (i.e. faster than exponential). — , Feb 14 '14 at 19:41
I've been referring to functions having the form $ \ f(x)^{g(x)} \ $ as "generalized exponential functions" in teaching courses, for lack of a better name. I can't say I've run across a commonly-used term for these... — colormegone, Feb 14 '14 at 19:55

score 8 · Accepted Answer · answered Feb 14 '14 at 18:45

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

answered Feb 14 '14 at 18:45

Ben Grossmann

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DeepSea · Answer 2 · 2016-03-26T15:58:54.657

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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.

edited Mar 26 '16 at 15:58

answered Feb 14 '14 at 18:41

DeepSea

77,651

Is there a good way to describe its growth? – Paul Feb 14 '14 at 18:43
@E-theory Would you please be so kind to define a square? – Michael Hoppe Feb 14 '14 at 19:02
@E-theory So won't call $x\mapsto\left(\frac12\right)^x$ an exponential map? – Michael Hoppe Feb 14 '14 at 19:05
Please ignore my last comment, instead I would like to ask how do you call $x\mapsto2^{\sin(x)}$. – Michael Hoppe Feb 14 '14 at 19:18

Jakub Skop · Answer 3 · 2019-07-12T04:23:58.347

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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

edited Jul 12 '19 at 04:23

answered Jul 08 '19 at 15:46

Jakub Skop

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Also, why the downvote - if there is anything wrong please let me know. – Jakub Skop Jul 10 '19 at 17:02

Michael Hoppe · Answer 4 · 2014-02-14T19:12:22.687

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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.

edited Feb 14 '14 at 19:12

answered Feb 14 '14 at 18:47

Michael Hoppe

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I don't understand. Are you trying to find a name for $x^x$? It may be considered to be a power or an exponentiation as $x\cdot x=x^2$ may be considered as a product or as a power. – Michael Hoppe Feb 14 '14 at 18:59
Why was the answer downvoted? – Michael Hoppe Feb 15 '14 at 10:41
I don't understand why people downvote sometimes. I thought it was an interesting approach. – Richard Mar 26 '16 at 09:29
I think the downvote might have been because of the comment practically useless. There is rather a lot of research into power towers for something that is useless. – Jesse Chisholm Dec 27 '21 at 02:08

Is $x^x$ an exponential function?

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