Multiplicative Modulus Function Explained

[lolgarithms]

"multiplicative modulus"

I have found a multiplicative modulus function, an analogue of absolute value for multiplication. Not a groundbreaking one, I'm sure someone has thought about this before. More specifically it is a function that satisfies:

$$\operatorname{MM}(1) = 1$$
$$\operatorname{MM}(x^{-1}) = \operatorname{MM}(x)$$
$$| \log{x} | = \log{(\operatorname{MM}(x))} \, \forall x \in \mathbb{R}^+$$
$$\operatorname{MM}(x) = x \, \forall x \ge 1$$
$$\operatorname{MM}^{-1} (x) = x^{\pm 1}$$

Why isn't this function widely used in mathematics? is it because it can be replaced with $$e^{|\ln{x}|}$$ ?

 

[matt grime]

May I simplify?

We define a function on the strictly positive real numbers:f(x)=x if x >=1,
f(x)=1/x otherwise.

It probably isn't widely used because no one has needed to use it.