# How would describe the proportionality of $y={k*a^x}$? Is it "exponentially proportional"?

Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table.

So I understand "inversely proportional" to mean the frequency of a word of a given rank $r$ is $f(w) = K / r(w)$ (where $K$ is a constant, the frequency of the top-ranked word, and $r(w)$ is the rank of word $w$).

However, in my corpus I found $f(w) = K * 0.9^{r(w)}$. I'm pretty certain this can't be said to be "inversely proportional".

Here is my question: How would you describes in words the correspondence represented by $f(w) = K * 0.9^{r(w)}$? Is there a good phrasing for this? Is it "frequency is exponentially proportional to rank"? Wikipedia seems to suggest so, but because $a < 1$ values shrink instead of exploding like "exponentially proportional" suggests to my ears...

$$K \cdot 0.9^{r(\omega)} = K \cdot \left( \frac{10}{9} \right)^{- r(\omega)}$$
It gets smaller because the base is less than one. As $r(\omega)$ increases, $f(\omega)$ decays exponentially, and as $r(\omega)$ decreases, $f(\omega)$ grows exponentially.
• Ah, so I'm understanding that I should describe it as "frequency decays exponentially proportional to rank", yes?