# Shannons Entropy

I am reading the proof of Shannons Entropy in his paper of 1948. (http://math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf)

In the proof (Aprendix 2, page 29), I do not understand why

$|\frac{A(t)}{A(s)}-\frac{\log t}{\log s}|<2\epsilon$

implies that

$A(t)=K \log t$

Where does the $K$ came from?

## 1 Answer

Since $\epsilon$ is arbitrarily small, Shannon has proved that

$\frac{A(t)}{A(s)} = \frac{\log(t)}{\log(s)} \quad \forall s,t \in \mathbb{Z}^+$

Re-arranging this identity gives

$\frac{A(t)}{\log(t)} = \frac{A(s)}{\log(s)} \quad \forall s,t \in \mathbb{Z}^+$

If we call this ratio $K$ then we have

$\frac{A(t)}{\log(t)} = K \quad \forall t \in \mathbb{Z}^+$

$\Rightarrow A(t) = K \log(t)\quad \forall t \in \mathbb{Z}^+$

• $s$ and $t$ are arbitrary positive integers so it doesn't matter whether we use $s$ or $t$ - it is a bound variable. I have made an edit to clarify this. – gandalf61 Jun 4 '18 at 14:25
• Thank you very much! I got it! :) – Patricio Jun 7 '18 at 14:07