As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via $\exp:\mathbb R\to\mathbb R^{+}$ I get the norm $\|\cdot\|_{\log}$ on $(\mathbb R^{+}, \cdot)$ which is defined by $$\|a\|_{\log}=|\log(a)|$$ and the distance $$\operatorname{dist}_{\log}(a,b) = \left|\log\left(\frac ab\right)\right| = |\log(a)-\log(b)|$$

My Question: Is there a name for the norm $\|\cdot\|_{\log}$ or for the error/distance $\operatorname{dist}_{\log}(a,b)$? Can you point me to a textbook/paper, where properties of this norm/distance are discussed?

Reason for my Question: I'm interested in interval arithmetic where intervals are defined like the following:

$$\begin{align} [a]_\epsilon &= \left\{ y \in \mathbb R^{+}: \left\|\frac{y}{a}\right\|_{\log} = |\log(y)-\log(a)| < \epsilon \right\} \\ &= \left[ae^{-\epsilon};ae^{\epsilon}\right] \end{align}$$

Unfortunately I do not now how the norm $\|\cdot\|_{\log}$ nor the error $|\log(y)-\log(a)|$ is called. So I do not know for what I shall look for...


The metric $$ \mathrm{dist}(x,y) = |\ln(x) - \ln(y)| = \left| \ln\left( \frac{x}{y} \right) \right| $$ is frequently used in compositional data analysis, where it was introduced by J. Aitchison. It is known as the log-ratio distance.

It is far less well knwon that this metric also appears in the log-normal distribution. Conventionally, the probability density function of the log-normal distribution is written down as $$ f_\mathcal{LN}(x \mid \mu,\sigma^2) = \dfrac{1}{\sqrt{2\pi} \sigma x} \exp\left( -\dfrac{ \left(\ln(x) - \mu \right)^2 }{2\sigma^2} \right). $$ However, certain conceptual relationships, such as the involvement of the squared distance of observations from the mean, are obscured in this expression. In fact, if parameterized with the geometric mean $m =\mathrm{e}^\mu$, the probability density function becomes $$ \begin{array}{rcl} f_\mathcal{LN}(x \mid m,\sigma^2) &=& \dfrac{1}{\sqrt{2\pi\sigma^2 x^2}} \exp\left( -\dfrac{ \left| \ln(x) - \ln(m) \right|^2 }{2\sigma^2} \right) \\ &=& \dfrac{1}{\sqrt{2\pi\sigma^2 x^2}} \exp\left( -\dfrac{ \mathrm{dist}^2(x,m) }{2\sigma^2} \right). \end{array} $$


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