Scale Invariance of the Distribution of First Digits

Question

Why is the distribution of first digits scale invariant? That is, if we multiply all our numbers by an arbitrary constant then why does the distribution of first digit frequencies remain unchanged. This is related to Benford's Law. I was reading through https://plus.maths.org/content/looking-out-number-one#:~:text=Using%20this%20reasoning%2C%20Pinkham%20went,Benford's%20Law%20really%20does%20exist. and was confused when going through the first two paragraphs in the section "Pinning down Scale Invariance"

Answer 1

You need to look at what is assumed and what is derived. I believe that we assume or discover empirically that the distribution is scale invariant. We then derive Benford's law from this. The simple idea is that if the distribution is scale invariant, we can multiply everything by a scale factor. If we choose that factor to be $5$, we take the points that had a leading digit $1$, meaning they were in the interval $[1,2)$ and spread them among the interval $[5,10)$ with leading digits $5$ through $9$. Looking at this in detail, we find that the fraction of points between $a$ and $b$ is $\log b - \log a$. Specializing that to leading digits gives Benford's law.