A Clever Formula for Logarithm

A Clever Formula for Logarithm

One of the best mathematical tools ever developed is the logarithm of a number. It has been used extensively in the past for the simplification of lengthy arithmetic calculations. The standard way of using the technique is via tables of common logarithms. The method of using these tables is well known and has been in use for decades. In this article, we will talk about a lesser known method of finding log of a number without using the tables.

This doesn’t really work, not for most uses of logarithms. This article could really use a reference or a derivation of the formula, at least something to explain where it came from and how the magic number 0.4343 was computed.

Here are a couple of Wolfram Alpha plots of this method (with b=7) vs. the true value of the log function:

http://plot 1 to 1000
http://plot 0 to 1

They show how rapidly this formula diverges from the true value.

A quick shorthand for base-10 log is to simply count the number of digits in a number, this always will be accurate to within 1. For numbers less than 0, count the number of zeros to the right of the decimal point and multiply by -1 (e.g. 0.001 -> 2).

Hi @stephen-marsh .Thanks for the comments.

Let me explain a bit more about how the formula works:

  1. The choice of b is not arbitrary. It has be close to a. For example, if you need to find the value of log(6.45), you may choose b = 6.5 or 6 or 7, but not 2 or 3 or 4. That is because these numbers are not close to 6.45 compared to 6 or 7. So, your results will not be correct. This is the reason your plots show a divergent behavior.

  2. The magic number 0.4343 is simply the log10(e). Its just a conversion factor from base-e to base-10.

  3. I did not discuss the derivation in the article as it involved a bit of calculus. I was trying to keep the content less rigorous. But, if more people are interested to know the derivation, I will put up another article describing it in detail. Otherwise you can send me an email if you want a complete derivation.