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

**log**_{e} 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 e**^{y} = a. Hence ln(e) = 1.To understand it more clearly, Lets check the table of power of 2.

2^{0} = 1 | 2^{9} = 512 |

2^{1} = 2 | 2^{10 }= 1024 |

2^{2} = 4 | 2^{11} = 2048 |

2^{3} = 8 | 2^{12} = 4096 |

2^{4} = 16 | 2^{13} = 8192 |

2^{5} = 32 | 2^{14} = 16384 |

2^{6} = 64^{} | 2^{15} = 32768 |

2^{7} = 128 | 2^{16} = 65536 |

2^{8} = 256 | 2^{17} = 131072 |

Let’s see how multiplication can be done easily.

Multiply 128 and 512.

From table 128 stands for 2

^{7}And 512 stands for 2

^{9}So 128 x 512 can be written as 2

^{7 }x 2

^{9}By index rule, 2

^{7 }x 2

^{9} = 2

^{(7+9)} = 2

^{16}So the required product is the number in the table equivalent to 2

^{16} 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 2

^{14}And 256 stands for 2

^{8}So,

$\frac{16384}{256}$ is same as

$\frac{2^{14}}{2^{8}}$By index rule

$\frac{2^{14}}{2^{8}}$ = 2

^{(14 - 8)} = 2

^{6}So the required product is the number in the table equivalent to 2

^{6} 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