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

In summary, the conversation discusses the use of the terms "power", "index", and "exponent" and their differences in meaning. It is mentioned that "exponent" is a more formal term for "power" and that "index" is used differently in British and American English. The conversation also touches on the use of the term "exponential growth" and what term to use for other types of growth described by functions such as polynomials. It is clarified that "power" and "exponent" are used to describe the operation of raising one number to the power of another, while "exponentiation" is reserved for raising the number e or other specific numbers to a power. The conversation also notes that "exponential growth
  • #1
Aeneas
27
0
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
 
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  • #2


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.
 
  • #3


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).
 
  • #4


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).
 
  • #5


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?
 
  • #6


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
 

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

What is the difference between 'power', 'index' and 'exponent'?

The terms 'power', 'index', and 'exponent' are often used interchangeably, but they have slightly different meanings in mathematics. 'Power' refers to the result of raising a number to a certain value, while 'index' is the value that is used to raise the number. 'Exponent' is another word for 'index', but it is often used in the context of scientific notation. For example, in the expression 23, 3 is the index or exponent, and 8 is the power.

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

While these terms are closely related, they are not exact synonyms. As mentioned, 'power' and 'exponent' are often used interchangeably, but 'index' is a more specific term that refers to the value used to raise a number. Additionally, 'power' can have other meanings in different contexts, such as electrical power or political power, while 'index' and 'exponent' are strictly mathematical terms.

How are 'power', 'index', and 'exponent' used in algebra?

In algebra, 'power' and 'exponent' are used to represent repeated multiplication. For example, x3 means x multiplied by itself three times. 'Index' is often used in the context of logarithms, where it represents the power to which a base number is raised to equal a given number. For example, log28 = 3, where 2 is the base, 8 is the number, and 3 is the index or exponent.

Can 'power', 'index', and 'exponent' be negative?

Yes, all three terms can have negative values. In algebra, a negative exponent indicates the reciprocal of the number raised to the positive exponent. For example, x-3 is the same as 1/x3. In logarithms, a negative index indicates that the base number is less than 1. For example, log20.5 = -1, where 2 is the base, 0.5 is the number, and -1 is the index or exponent.

How do 'power', 'index', and 'exponent' relate to each other in mathematical operations?

In mathematical operations involving exponents, the power can be thought of as the product of the base number multiplied by itself the number of times indicated by the index or exponent. For example, 23 = 2 x 2 x 2 = 8. In addition, when raising a power to another power, the indices are multiplied together. For example, (23)2 = 26 = 64.

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