Are 'power', 'index' and 'exponent' exact synonyms

[Aeneas]

Can you please help me sort out my terminology?

Are 'power', 'index' and 'exponent' exact synonyms, even thogh they tend to be used in different contexts? If a^x gives 'exponential growth' is the growth described by x^a also properly called 'exponential'? If not, what is it called?

Thanks,

Aeneas

 

[HallsofIvy]

I would consider "power" and "exponent" to be basically the same- "exponent" being a little more formal than "power". Our British friends use "index" to mean "exponent" but we Americans do not. To us an "index" is simply a "label" (as on a vector or tensor) and can be either a superscript of a subscript.

"Exponential growth" on the other hand refers to the "exponential function", e^x or variations on that such as a^x= e^{x ln(a)}. Something like x^a is a "polynomial function" if a is a positive integer, a "rational function" if a is a negative integer, a "radical function" if a is a fraction, and a "transcendental function" if a is irrational.

 

[arildno]

Note:

Often, in modelling, to utilize a function:
$$f(x)=Cx^{a}$$
is called to use a "power law". (C, a constants to be empirically determined).

 

[uart]

Yes that's the way I refer to them.

$f(x) = a^x$ : an exponential.

$f(x) = x^a$ : a power (of x).

 

[Aeneas]

Thanks for those replies. Can you use "exponentiation" as a noun, to go with "addition" and "multiplication" for example, to generally describe the general process of raising one number to the power of another, then, or should it be reserved for raising e or some other number to the power of x?

Also, the phrase "exponential growth" is a common one, but what would you put in the bracket in "( ) growth" if the growth was described by, say, a polynomial function?

 

[Elucidus]

These are the distinctions as I know them:

A "power" is an operation also known as exponentiation, as in the third power of 2 is 8.

The "exponent" is the argument in the superscript of a power - then n in aⁿ. It is also the "index" of the power in the same way as n is the index of the radical $$\sqrt[n]{a}$$.

For a constant:

$f(x) = a^x$ is an exponential function.

$g(x) = x^a$ is a power function.

I hope this helps.

--Elucidus