Is $d(x,y)=|\log(x/y)|$ the only metric that makes $(\mathbb{R}_+,d)$ complete?

Question

Let $\mathbb{R}_+$ be the set of strictly positive real numbers. Consider the following distance between elements of $\mathbb{R}_+$ $$ d(x,y)=\left|\log\left(\frac{x}{y}\right)\right|. $$

It is easy to prove that this distance makes the metric space $(\mathbb{R}_+,d)$ complete. But is $d$ the unique distance with this property?

Answer 1

Take any continuous, injective function $f:\Bbb R_+\to \Bbb R$ such that $\lim_{x\to 0^+}f(x) = -\infty$ and $\lim_{x\to \infty}f(x) = \infty$ (or vice versa). Then $d(x, y) = |f(x) - f(y)|$ will give a complete metric space. In fact, it will be isometric, via $f$, to $\Bbb R$ with the standard metric.

Your example uses $f(x) = \log(x)$, and orangeskid in the comment above uses $f(x) = x - \frac1x$.