# How to indicate that a function is exponential?

This may be a very specific question and honestly I am still unsure whether I should be asking this here or in the Math Exchange. Maybe it is something very specific to mathematics, I don't know.

So the question is the following: suppose I have a function

f(x,n) = x^n (meaning, x to the power of n, both n and x are variables)

I would like to write a sentence like this: "Function f is exponential [OPTION] n".

Now, [OPTION] could be:

1. with respect to
2. on
3. in

Or maybe some other experission/preposition. I guess (1) is the safest option (?) but I have been writing (2) for some time now until a colleague called my attention to it and made me realize I have no idea what is the correct manner of writing the sentence.

Thus, what would be the correct way of writing the sentence?

• This should probably have been asked on SO Math. As an Anglophone, I'd expect This function is exponential with respect to N, but in Google Books I just found Thus the isoefficiency function is exponential in N. I see no relevant matches for This function is exponential on... Feb 22 at 12:40
• Not "on", as that would usually be used to denote (a subset of) the domain of the function ("This function is real-valued on the integers"). Both "in" and "with respect to" are used in the sense you want; I prefer "in" because it's shorter and simpler. Feb 22 at 12:44
• 'Growth is exponential with respect to time' is fine, but N = Nₒ ͭ (I can't manage the t-1 superscript that's required) is the way this would be written with variables. // You've chosen a two-variable function; n is usually a constant. z = f(x,y) = x^y would be more regular. This Mathematics.SE question covers this. The writer labels this 'linear in x, exponential in y', but gives no supporting reference. Feb 22 at 12:47
• We have a fairly high percentage of mathematicians, programmers, and such here, so you might get lucky! Feb 22 at 12:55
• If "n" is a variable then "with respect to the power" would be correct. Feb 22 at 15:19

The super-standard expression is

exponential in n,

provided you understand that in many contexts, it includes a broader range of functions than just the one you've written. Basically, exponential in n means that the 'dominant scaling' with n is exponential. For example, for every one of the following functions, there are fields where it would be described as 'exponential in n' (or in k, or in d… , as the case may be):

2n/log n     (' exponential in n ')
In Nielsen, Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information, 2000 (link)

2k+1 k log k    (' exponential in k ')
In Advances in Neural Information Processing Systems 15: Proceedings of the 2002 Conference (link)

qd2    (' exponential in d '; yes, they say d, not d2)
In Aspects of Complexity: Minicourses in Algorithmics, Complexity and Computational Algebra: Mathematics Workshop, Kaikoura, January 7-15, 2000 (link)

It is not an accident that all these are in complexity-related fields. When judging whether a quantity is 'exponential' in a parameter, all that the practitioners in these fields care about is that the parameter enters some exponent, and that the way it enters the exponent is significantly faster than logarithmic. (After all, the logarithm can 'cancel out' the exponential: eln x = x.)

Having said that, in some fields, the usage is stricter. For example, the 2n log n scaling is referred to as super-exponential here, while any scaling 2o(n) is called sub-exponential here (where the 'little o notation' is being used: f(n) = o(n) as n → ∞ means that that f(n)/n → 0 in that limit). No doubt, there are many fields where 'y is exponential in n' could only mean that y = a bn for some a and positive b. But since we don't know the relevant field (you may!), we can't exclude the possibility that the meaning of 'exponential in (a parameter)' is as broad as in the examples I gave above.

If you want to forestall this broad interpretation, you will have to rephrase and use a longer description, e.g.

y is proportional to the exponential function of a constant multiple of n.

• First example, see link. It uses "big O" operator notation. The phrase "exponential in n" is a reference to the resultant Big O operation , not the function itself. This is important - bad example. Feb 23 at 2:56
• Third example, see link "the order of the general linear group is exponential in d" - not the function, the function's linear group. Bad example. Please find an example where "exponential in n" has been applied to x^n. This is canonical stuff. Feb 23 at 3:10
• @PhilSweet Please note that the OP is using the phrasing Function f is exponential … not Function f is an exponential …. This is the same construction as in the third example. For a related reason, I disagree that the first example is inappropriate. Without an article, one assumes that we are talking about scaling in all these examples, including the OP's, so indeed the big O notation is relevant. Feb 23 at 3:17
• Fine, but I am taking the exact opposite approach and viewing this as a strict application to x^n, and nothing other than that. It is a question about canonical monomials. Feb 23 at 3:22
• @PhilSweet And you have every right to your viewpoint… Feb 23 at 4:42

"Function f is exponential [OPTION] n".

Function f is exponential base x, exponent n.

or just

Function f is exponential with exponent n.

Note: You cannot simply state "wrt n" because that would be ambiguous for a two argument function.

• Would you say it is ambiguous even if I present the function where "n" is clearly the exponent? Feb 22 at 12:57
• To answer that question, we would need to see a complete actual paragraph (not an explanation), where you 'present' the function and then later describe it using your desired phrase. Context makes a big difference in such matters. Feb 22 at 13:09
• Just saying "x^n is exponential with respect to n" sounds fine to me, since, in my book, x^n is not exponential with respect to x; it's a power function with respect to x. Feb 22 at 13:20
• @Tanner Swett - Which book is that ;-) - Can you give evidence from the literature? If you are writing maths it is of the utmost importance to avoid ambiguities. Why risk it for the sake of an extra word? Feb 22 at 14:07
• @chasly-supportsMonica: There do seem to be plenty of examples in Google Books; there are even more for "exponential in n". I don't recall ever seeing "(function) is exponential base x" without setting off the base somehow (eg "exponential, base x, ..." or "exponential (base x)" or "exponential with base x" etc); do you have an example of that? Feb 22 at 14:29

It's not an exponentional function.

Using normal variable attributes, x is a real variable and n is an integer variable. So x^n is a monomial of x raised to n, or more formally, x^n is a monomial of x of degree n.  Wikipedia: monomial

<Can we please get mathjax support here on EL&U!!!>

A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, {\displaystyle x^{2}yz^{3}=xxyzzz}{\displaystyle x^{2}yz^{3}=xxyzzz} is a monomial. The constant 1 is a monomial, being equal to the empty product and to x0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power xn of x, with n a positive integer. If several variables are considered, say, {\displaystyle x,y,z,}x,y,z, then each can be given an exponent, so that any monomial is of the form {\displaystyle x^{a}y^{b}z^{c}}x^{a}y^{b}z^{c} with {\displaystyle a,b,c}a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1).