# Easiest way to find the log of any number without tables

Well, it is often needed to get the logarithm of a number without log tables.

So, it is advisable to know a few common techniques which may be useful in finding the logarithm of many numbers (logarithm to the base 10).

Below I have mentioned some useful ways to find the logarithm of numbers :

1.) log 2 = 0.3010

2.) log 3 = 0.4771

3.) log 7 = 0.8451

4.) log e = 0.693

5,) Learn the above 4 logarithms. They will be useful in computing the logarithm of other numbers that are frequently required in various competitive examinations

6.) log (ab) = log a + log b -> first logarithm identity

7.) log (a/b) = log a – log b -> second logarithm identity

8.) log (a^b) = b loga -> third logarithm identity

9.) now to compute the logarithm of many other numbers , we can use these identities along with the 4 standard algorithms mentioned above.

Let’s see some examples :

suppose you need to find log 5

now log 5 = log (10 / 2) = log 10 – log 2 (using second logarithm identity)

now we know that log 10 = 1

and log 2 = 0.3010

so log 5 = log 10 – log 2 = 1 – 0.3010 = 0.6990

this way, we were able to find log 5

now let us take more examples of computing log

suppose you are asked to find log 12

now log 12 = log (3 * 4) = log 3 + log 4 (using first logarithm identity)

now we directly know log 3 = 0.4771

but we need to calculate log 4

log 4 = log (2 * 2) = log 2 + log 2 (using first logarithm identity) = 2 log 2

alternatively, log 4 = log (2 ^ 2) = 2 log 2 (using third logarithm identity)

this way, log 12 = log 3 + 2 log 2 = 0.4771 + 2 * 0.3010 = 0.4771 + 0.6020 = 1.0791

find this using a calculator

you will conclude that our answer is correct upto 4 decimal places which is good enough for most competitive exams

using these small techniques, you can find the logarithm of a large number of numbers

however, there are a few limitations

logarithm of some numbers cannot be found using this method

for example, you cannot find log 11 using this technique

think why ?