Which is actually exponential?

Question

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?

Answer 1

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.

Answer 2

$n^x$ 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.

Answer 3

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.

Answer 4

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.

Answer 5

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!