derivation of Hartley function
We want to show that the Hartley function ${\mathrm{log}}_{2}(n)$ is the only function mapping natural numbers^{} to real numbers that

1.
$H(mn)=H(m)+H(n)$ (),

2.
$H(m)\le H(m+1)$ (monotonicity), and

3.
$H(2)=1$ (normalization).
Let $f$ be a function on positive integers that satisfies the above three properties. Using the additive^{} property, it is easy to see that the value of $f(1)$ must be zero. So we want to show that $f(n)={\mathrm{log}}_{2}(n)$ for all integers $n\ge 2$.
From the additive property, we can show that for any integer $n$ and $k$,
$$f({n}^{k})=kf(n).$$  (1) 
Let $a>2$ be an integer. Let $t$ be any positive integer. There is a unique integer $s$ determined by
$$ 
Therefore,
$$ 
and
$$ 
On the other hand, by monotonicity,
$$f({a}^{s})\le f({2}^{t})\le f({a}^{s+1}).$$ 
Using Equation (1) and $f(2)=1$, we get
$$sf(a)\le t\le (s+1)f(a),$$ 
and
$$\frac{s}{t}\le \frac{1}{f(a)}\le \frac{s+1}{t}.$$ 
Hence,
$$\left\frac{1}{f(a)}\frac{1}{{\mathrm{log}}_{2}(a)}\right\le \frac{1}{t}.$$ 
Since $t$ can be arbitrarily large, the difference^{} on the left hand of the above inequality must be zero,
$$f(a)={\mathrm{log}}_{2}(a).$$ 
Title  derivation of Hartley function 

Canonical name  DerivationOfHartleyFunction 
Date of creation  20130322 14:32:15 
Last modified on  20130322 14:32:15 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  11 
Author  Mathprof (13753) 
Entry type  Derivation 
Classification  msc 94A17 