A logarithmic system in which the neper constant ($e$) is used as a base, is called the natural logarithm.

John Napier firstly introduced the logarithmic system by using an irrational mathematical constant $e$ as base of the logarithm to express any number into the multiplying terms of this constant.

He was developed it from natural mathematical procedure. Hence, it is called as the natural logarithms. Due to use of Napier’s constant in developing this logarithmic system, it is also called as Napier logarithms.

John Napier defined the value of neper’s constant is an irrational number $(e)$ and it is equal to $2.71828182845904523536028747135266249775724709369995…$. The approximate value of e is 2.71828.

It’s equal to neither

$1$nor negative real numbers but belongs to positive real numbers.

$e=2.71828\ne 1$

and

$\notin {R}^{\u2013}$but

$\in {R}^{+}$.

So, Neper’s constant is eligible to be the base of logarithmic system as per the principle rule of the logarithmic system.

Natural logarithmic system is used to know how many times napier constant is multiplied to itself to form a particular number as product.

$150$ is a number and express it in multiplicative terms of Napier’s constant $e$.

$150 = e \times e \times e\times e \times e \times e^{\displaystyle 0.010635294…}$

$\Rightarrow 150 = e^{\displaystyle 5.010635294…}$

The number $150$ is obtained if the neper’s constant is multiplied to itself $5.010635294…$ times.

Actually, it is impossible to us to express any number into multiplicative terms of $e$ but it is possible to know the number of multiplying terms of e to obtain any number with natural logarithmic system.

$\log_{\displaystyle e} 150 = 5.010635294…$

Natural logarithmic system can be expressed in mathematical form by using Algebra.

Assume, a number is

$x$. The number of multiplicative factors of

$e$is

$y$to form the number

$x$as product. Natural logarithm is written in algebraic form in mathematics in two different styles.

$y={\mathrm{log}}_{e}x$

(or)

$y=lnx$The relation between exponentiation system and natural logarithmic system is written in mathematics as follows.

${e}^{y}=x\iff {log}_{e}x=y$

It can also be written in simple form.

${e}^{y}=x\iff lnx=y$

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