# Understanding the use of logarithms in the TF-IDF logarithm

https://en.wikipedia.org/wiki/Tf%E2%80%93idf#Definition

But I cannot seem to understand exactly why the formula was constructed the way it is.

## What I do Understand:

iDF should at some level measure how frequently a term S appears in each of the documents, decreasing in value as the term appears more frequently.

From that perspective

$$iDF(S) = \frac{\# \text{ of Documents}}{\# \text{ of Documents containing S}}$$

Furthermore term frequency can be rightly described as

$$tf(S,D) = \frac{\# \ \text{of Occurrences of S in document D}}{\# \ \text{maximum number of occurrences for any string Q in document D}}$$

So then the measure

$$iDF(S) \times tf(S,D)$$

is in some way proportional to how frequently a term appears in a given document, and how unique that term is over the set of documents.

## What I don't understand

But the formula given describes it as

$$\left( \log(iDF(S)) \right) \left( \frac{1}{2} + \log(\frac{1}{2} tf(S,D)) \right)$$

I wish to understand the need for the logarithms described in the definition. Like, why are they there? What aspect do they emphasize?

The aspect emphasised is that the relevance of a term or a document does not increase proportionally with term (or document) frequency. Using a sub-linear function therefore helps dumped down this effect. To that extend the influence of very large or very small values (e.g. very rare words) is also amortised. Finally as most people intuitively perceive scoring functions to be somewhat additive using logarithms will make probability of different independent terms from $P(A, B) = P(A) \, P(B)$ to look more like $\log(P(A,B)) = \log(P(A)) + \log(P(B))$.
• Thank you for you comments gentlemen (and special thanks Alexey for the correcting the \log, I constantly forget them); +1 to both. I have seen the Robertson paper and contemplated adding it; it is a really good read, I will add it in the main body. – usεr11852 Jul 16 '15 at 9:59
• I want to know why "maximum number of occurrences for any string Q in document D" is used instead of number of occurrences for all strings in document D. Why do we want the count of the most common word instead of the count of all words? – Xeoncross Oct 12 '19 at 16:30