Estimating Logarithms

Summary

While reading through the fantastic book The Lost Art of Logarithms by Charles Petzold I was nerd-sniped by a simple method of estimating the logarithm of any number base 10. According to the book, it was developed by John Napier (the father of the logarithm) about 1615. In french the natural logarithm is also called “le logarithm népérien” in reference to the mathematician. The Method We note that due to the nature of the logarithm (always referring to base 10 from here one out), the logarithm of any number NNN is approximately equal to the number of digits of NNN minus one. This is quite easy to see when thinking about numbers between 100 and 1000 for example: 100≤N<1000 log⁡(100)≤log⁡(N)<log⁡(100) 2≤log⁡(N)<3\begin{align} 100 &\leq N < 1000 \\\ \log(100) &\leq \log(N) < \log(100) \\\ 2 &\leq \log(N) < 3 \end{align}100 log(100) 2​≤N<1000≤log(N)<log(100)≤log(N)<3​​ This approximation by itself might seem useless at first: knowing that the logarithm of 5 is between 0 and 1 is pointless. But in combination with the following property of logarithms: log⁡(ab)=b⋅log⁡(a)\log(a ^ b) = b \cdot \log(a)log(ab)=b⋅log(a) We can calculate the logarithm of any number with arbitrary precision using the following this algorithm. We note #N\#N#N as the number of digits of N minus one. log⁡(N)≈#N log⁡(N10)≈#(N10) 10⋅log⁡(N)≈#(N10) log⁡(N)≈#(N10)10\begin{align} \log(N) &\approx \# N \\\ \log(N ^{10}) &\approx \#({N^{10}}) \\\ 10 \cdot \log(N) &\approx \#({N^{10}}) \\\ \log(N) &\approx \frac{\#({N^{10}})}{10} \end{align}log(N) log(N10) 10⋅log(N) log(N)​≈#N≈#(N10)≈#(N10)≈10#(N10)​​​ Increase the exponent from 100 to 1000 and you’ve added another digit of precision. Henry Briggs used this method to compute the logarithms of 2 and 7 to the 14th digit. Calculating 210142^{10^{14}}21014 must have been quite a task. He probably rounded the values a lot before multiplying, as there is no use to the first few digits at those scales. Simply exchanging the complexity of calculating the logarith...