We want to show that the Hartley function $\log_{2}(n)$ is the only function mapping natural numbers to real numbers that satisfies

1. 1.

   $H(mn)=H(m)+H(n)$ (additivity),

2. 2.

   $H(m)\leq H(m+1)$ (monotonicity), and

3. 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 $f(n)=\log_{2}(n)$ for all integers $n\geq 2$.

From the additive property, we can show that for any integer $n$ and $k$,

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,

Using Equation (1) and $f(2)=1$, we get

and

Hence,

Since $t$ can be arbitrarily large, the difference on the left hand side of the above inequality must be zero,