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?

Solution 1:

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)}

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

q^d2    (' 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: e^{ln x} = x.)

Having said that, in some fields, the usage is stricter. For example, the 2^{n log n} scaling is referred to as super-exponential here, while any scaling 2^o(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.