# 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** up to 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 ?

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Hope this will help you in **competitive exams**

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