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.

We choose to work with base 10

Here, a and b are such that ∣a − b∣ is as small as possible and b is chosen according to the list above. The accuracy of the result depends on smallness of ∣a − b∣.

Let us try to evaluate the value of log(2.7). So, we have a = 2.7 and we choose b = 2.8. We can plug these values in the master formula to get

The values of log(2) and log(7) are already known(memorized!) from the list above, so we have

The exact value of log(2.7) = 0.4314 and the error in the result is just 0.03%

For numbers less than 2, multiply and divide with an appropriate factor to bring it close to 5 or 7 in order to minimize errors. For e.g.

Now, the master formula can be used as before.