Before going into Natural Logarithm concept,Let's look at what is logarithm and why it is introduced. It is a fact that when we compare the operations, addition and subtraction are very easy compared to multiplication and division.
So logarithm is a method that is used to convert multiplication and division to addition and subtraction respectively. This was invented long before the use of calculators.
.

## Natural Logarithm Definition

The natural logarithm is the logarithm to the base ‘e'.
‘e' is an irrational number approximately equal to 2.7182.....The natural logarithms are sometimes called as the Napier Logarithms. Natural logarithm is defined for all real numbers and they can also be defined for complex numbers. E is the Euler's number.
The natural logarithm of a number x is written as loge x or simply as ln x.
Many natural phenomena like the population growth, radioactive decay, certain bacterial growth can be defined using logarithms to the base e i.e. natural logarithms. (This may one of the reasons why the logarithm to the base e are called natural logarithms)

Natural logarithms can easily be defined by using Taylor's series. Ln (b) is the area under the graph of $\frac{1}{x}$ from 0 to b. If y = ln (a) then ey = a. Hence ln(e) = 1.To understand it more clearly, Lets check the table of power of 2.

 20 = 1 29 = 512 21 = 2 210 = 1024 22 = 4 211 = 2048 23 = 8 212 = 4096 24 = 16 213 = 8192 25 = 32 214 = 16384 26 = 64 215 = 32768 27 = 128 216 = 65536 28 = 256 217 = 131072

Let’s see how multiplication can be done easily.
Multiply 128 and 512.
From table 128 stands for 27
And 512 stands for 29
So 128 x 512 can be written as 27 x 29
By index rule, 27 x 29 = 2(7+9)
= 216
So the required product is the number in the table equivalent to 216 which is 65536
S0, 128 x 512= 65536
This answer was obtained without multiplication.

Division can also be done in a similar way.
Let’s take $\frac{16384}{256}$
From table 16384 stands for 214
And 256 stands for 28
So, $\frac{16384}{256}$ is same as $\frac{2^{14}}{2^{8}}$
By index rule $\frac{2^{14}}{2^{8}}$ = 2(14 - 8)
= 26
So the required product is the number in the table equivalent to 26 which is 64
So, $\frac{16384}{256}$ = 64

So, if the table was long enough to contain all powers of 2, the multiplication and division would have been a lot easy. But it has its limitations. In this term all real numbers are not present, so it’s not appropriate to use it in general

### Change of Base Formula

 Natural Logarithm Function Logarithms Even Natural Numbers Natural Number Logarithmic Differentiation Concentric Circles in Nature Derivative Natural Log Fibonacci Numbers and Nature Natural Counting Numbers Applications of Exponential and Logarithmic Functions Change of Base Formula for Logarithms Domain and Range of Logarithmic Functions
 Calculate Logarithm base 5 conversion Average Rate of Change Examples