# Learn How to Find Natural Logarithm (Log) Series of a Number - Tutorial

## How to Find Natural Logarithm (Log) Series of a Number - Definition, Formula, Example

##### Definition:

Logarithm of a number is an exponent to another number, (i.e,.) if n = ab then, b is log of n to the base a (loga n = b). They are, naperian logarithms and natural logarithms. Naperian logarithm is log to the base e and Natural logarithm is log to the base 10. We can convert Naperian Logarithm to common logarithms by means of the formula. log10n = logen x 0.43429448

#### Formula:

loge(1+x) = x - x2 / 2 + x3 / 3 - x4 / 4 + ... ∞ loge(1-x) = - x - x2 / 2 - x3 / 3 - x4 / 4 - ... ∞ loge((1+x) / (1-x)) = 2 (x + x3 / 3 + x5 / 5 + ... ∞)
##### Example :

Find log(1+3), log (1-5), log((1+4) / (1-4)) using logarithmic series

##### Solution :
###### Step 1:

Finding the value of log(1+3) loge(1+x) = x - x2 / 2 + x3 / 3 - x4 / 4 + ... ∞ log(1+3) = 3 - 32 / 2 + 33 / 3 - 34 / 4 + ... log(1+3) = 3 - (9/2) + (27 / 3) - (81 / 4) + ... log(1+3) = -1.0714e+22 (upto 50 values)

###### Step 2:

Finding the value of log(1-5) loge(1-x) = - x - x2 / 2 - x3 / 3 - x4 / 4 - ... ∞ log(1-5) = - 5 - 52 / 2 - 53 / 3 - 54 / 4 + ... log(1-5) = - 5 - (25 / 2) - ( 125 / 3) - ( 625 / 4 ) - ... log(1-5) = -2.1761e+22 (upto 50 values)

###### Step 3:

Finding the value of log((1+4)/(1-4)) loge((1+x) / (1-x)) = 2 (x + x3 / 3 + x5 / 5 + ... ∞) log((1+4) / (1-4)) = 2 (4 + 43 / 3 + 45 / 5 + ...) log((1+4) / (1-4)) = 2 (4 + (64 / 3) + ( 1024 / 5) + ...) log((1+4) / (1-4)) = 1.1047e+22 (upto 50 values)