# Constructing a completely additive increasing/decreasing function

Since additive functions have multiple definitions, I should note that I am using this definition.

$f(x)$ is completely additive iff $f(x_1 \times x_2) = f(x_1) + f(x_2)$. There are several such functions: logarithms, number of prime factors, etc.

I was wondering if anyone knew of the existence of such functions that are either non-increasing or non-decreasing functions (and efficiently computable)?

So far, the logarithms appear to be the only such function I can think of. Unfortunately, logarithms are not computable in polynomial time (for the sake of computational complexity) for all rational numbers, due to their irrational range.

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