Counterfeit Coin problem with a twist?

We all know the counterfeit coin problem with 7 real coins & one fake one.

n coins = x weighings

2 coins = 1 weighing
3 coins = 1 weighing
4 coins = 2 weighings
5 coins = 2 weighings
6 coins = 2 weighings
7 coins = 2 weighings
8 coins = 2 weighings
9 coins = 2 weighings
10 coins = 3 weighings
11 coins = 3 weighings
12 coins = 3 weighings
etc.

Well I get the problem and I sort of have the answer to it.
n^2 = k & the amount of digits k has = x weighs

How do I explain that in one equation/math sentence?
like n^2 x 3^3 or something like that…

Help please. Thanxx =]
I know how to solve it.

I just need to know how to put that the number of digits n^2 has is the amount of weighings or x

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