I got an evaluation copy of The Best Writing on Mathematics 2021 yesterday. One article jumped out as I was skimming the table of contents: *A Zeroth Power Is Often a Logarithm Yearning to Be Free* by Sanjoy Mahajan. Great title.

There are quite a few theorems involving powers that have an exceptional case that involves a logarithm. The author opens with the example of finding the antiderivative of *x*^{n}. When *n* ≠ −1 the antiderivative is another power function, but when *n* = −1 it’s a logarithm.

Another example that the author mentions is that the limit of power means as the power goes to 0 is the geometric mean. i.e. the exponential of the mean of the logarithms of the arguments.

I tried to think of other examples where this pattern pops up, and I thought of a couple related to entropy.

*q*-logarithm entropy

The definition of *q*-logarithm entropy takes Mahajan’s idea and runs it backward, turning a logarithm into a power. As I wrote about here,

The natural logarithm is given by

and we can generalize this to the

q-logarithm by definingAnd so ln

_{1}= ln.

Then *q*-logarithm entropy is just Shannon entropy with natural logarithm replaced by *q*-logarithm.

## Rényi entropy

Quoting from this post,

If a discrete random variable

Xhasnpossible values, where theith outcome has probabilityp_{i}, then the Rényi entropy of order α is defined to befor 0 ≤ α ≤ ∞. In the case α = 1 or ∞ this expression means the limit as α approaches 1 or ∞ respectively.

…

When α = 1 we get the more familiar Shannon entropy:

In this case there’s already a logarithm in the definition, but it moves inside the parentheses in the limit.

And if you rewrite

*p*^{α}

as

*p* *p*^{α−1}

then as the exponent in *p*^{α−1} goes to zero, we have a logarithm yearning to be free.

Another simple one is the Box-Cox transformation.