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

  1. Sep 7, 2009 #1
    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
     
  2. jcsd
  3. Sep 7, 2009 #2

    HallsofIvy

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    Re: Terminology

    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", ex or variations on that such as ax= ex ln(a). Something like xa 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.
     
  4. Sep 7, 2009 #3

    arildno

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    Re: Terminology

    Note:

    Often, in modelling, to utilize a function:
    [tex]f(x)=Cx^{a}[/tex]
    is called to use a "power law". (C, a constants to be empirically determined).
     
  5. Sep 7, 2009 #4

    uart

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    Re: Terminology

    Yes that's the way I refer to them.

    [itex]f(x) = a^x [/itex] : an exponential.

    [itex] f(x) = x^a [/itex] : a power (of x).
     
  6. Sep 7, 2009 #5
    Re: Terminology

    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?
     
  7. Sep 7, 2009 #6
    Re: Terminology

    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 [tex]\sqrt[n]{a}[/tex].

    For a constant:

    [itex]f(x) = a^x[/itex] is an exponential function.

    [itex]g(x) = x^a[/itex] is a power function.

    I hope this helps.

    --Elucidus
     
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