# Which is actually exponential?

I've heard the term "exponential" applied to two sorts of functions:

$$n^x\text{, where n is a constant (e.g., 2^x)}$$

and

$$x^2$$

Which is really exponential, and what do I call the other one that is not exponential?

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Sorry by x^n, I really meant a specific x^n: x^2. Do the answers still apply? – George Newton Mar 28 '14 at 2:19
Yes, see my edit. – Alex Becker Mar 28 '14 at 2:29

The first is exponential. The second is polynomial.

Edit: $x^2$ is polynomial, but since $2$ is so small we have a special name for it as well: quadratic.

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Thank you very much. – George Newton Mar 28 '14 at 2:32
I find this only partly correct. Please see below for a more complete answer. – mirkastath Mar 28 '14 at 4:45

Although $p(x)=x^n$ is a specific polynomial, functions of this form are usually called power functions. Functions of the form $f(x)=n^x$ are usually called exponential functions.

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$n^x$ is an exponential function, $x^n$ is a polynomial if $n$ is an natural number (and zero if you don't consider $0$ a natural number), otherwise, it's some root of $x$ (e.g. $x^\frac{1}{2} = \sqrt{x}$ or $x^\frac{5}{3} = \sqrt[3]{x^5}$).

...now in $x^n$ if $n$ is an irrational number, then it's not exactly correct to say it's a root, but the irrational number can be approximated by a rational root.

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THE exponential function is $e^x$.

http://mathworld.wolfram.com/ExponentialFunction.html

But we often speak of "exponential growth", related to geometric growth, described by functions resembling n^x, in particular see here:

http://en.wikipedia.org/wiki/Exponential_growth

By the way: "Exponential growth" makes sense, because: if, for instance the rate of change dN/dt of a population (or number of radioactive nuclei in a sample and so on) is proportional to the population at any given time,

$dN(t) = r N(t) dt$ [In Wikipedia's example, $dt=1$ and $dN(t)=x(t+1)-x(t)$]

then $N(t) = N(0) e^{rt}$, i.e., $N(t)$ is the exponential function.

Edit 1: To conclude, applying the term "exponential function" in the cases you mention is either incorrect $(x^2)$ or a loose and inaccurate usage $(2^x)$.

Edit 2: Eqs. beautified using latex's $, thanks @Ruslan for the comment below Edit 3: My judgement on$2^x$being called exponential was likely too harsh, see comments below. Indeed,$a^x=e^{( ( \ln a) ) x }$is the exponential function! - I don't think it's fair to call using "exponential" for$2^x$"inaccurate". While$e^x$is indeed "The" exponential function, in many areas such as combinatorics or CS the term "exponential" is used much more broadly, e.g. the complexity class EXPTIME. – Alex Becker Mar 28 '14 at 4:51 @mirkastath You can enclose$\LaTeX$code in $s to get nicely rendered equations. – Ruslan Mar 28 '14 at 5:30
@AlexBecker You are probably right. Many people are taught already the term "exponential function" to apply to any $a^{bx}$. It appears even in Gradsteyn+Ryzhik as case 1.211 of exponential functions already, $a^x=\sum_{0}^{\infty} \frac{(x \ln{a})^k}{k!}.$ – mirkastath Mar 28 '14 at 5:58

For $n>0$ and $n\neq 1$, $f(x)=n^x$ is exponential.

For $n\in \{1,2,3,...\}$, $g(x)=x^n$ is polynomial.

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