From the

__introduction to logarithm__, we know that the value of a logarithm is meaningless if we do not mention the base.### Common Logarithm:

The logarithm with base 10 is called the common logarithm. This is also known as decimal logarithm.

For example, `log_{10} 5,` `log_{10} 7` are the common logarithms of 5 and 7 respectively.

**Solved Problems:**

**Ex 1:****Calculate the common logarithm of 10.**

__Solution:__The common logarithm of 10 is denoted by `log_{10} 10`

As we know that `log_a a=1`, we have `log_{10} 10=1`.

Thus the common logarithm of 10 is 1.

**Ex 2:****Find the common logarithm of 100.**

__Solution:__The common logarithm of 100 is denoted by `log_{10} 100`

By the power rule of logarithm `[log_a b^n=n log_a b]`,

we get that

`log_{10} 100=log_{10} 10^2=2 log_{10}{10=2 cdot 1=2`

Thus the common logarithm of 100 is 2.

Next, we discuss the natural logarithm.

**Natural Logarithm:**

The logarithm with base `e` is called the

**natural logarithm.**Here `e` is the well-known irrational number whose value is`e=sum_{n=0}^infty frac{1}{n!}`.

The natural logarithm of `M` is usually denoted as `ln M`. Note that `log Mneq ln M`.

**Solved Problems:**

**Ex 3:****Calculate the natural logarithm of `e`.**

__Solution:__We have to calculate `ln(e)`

We know that the natural logarithm has base `e`

Thus, `ln(e)=log_e e`

As `log_a a=1`, we deduce that `log_e e=1`.

Thus the natural logarithm of `e` is `1`.

That is, `ln(e)=1`

**Ex 4:****Find the natural logarithm of `-1`.**

__Solution:__We have to calculate `ln(-1)`

As the natural logarithm has base `e`, we have

`ln(-1)=log_e (-1)`

We know that `-1=e^{i pi}`

`therefore log_e (-1)=log_e e^{i pi}`

`Rightarrow log_e(-1)=log_e e^{i pi}`

`Rightarrow log_e(-1)=i pi quad` `[ because log_a a^b=b]`

So ` ln(-1)=i pi`.

Thus the natural logarithm of `-1` is `i pi`.

That is, `ln(-1)=i pi`

**Ex 5:****Calculate the natural logarithm of `i`.**

__Solution:__We have to calculate `ln i`

As the natural logarithm has base `e`, we have

`ln i=log_e i`

We know that `i=sqrt{-1}`

`therefore log_e i=log_e sqrt{-1}`

`Rightarrow log_e i=log_e (-1)^{1/2}`

`Rightarrow log_e i=1/2 log_e (-1) quad` `[ because log_a x^b=blog_a x]`

`Rightarrow log_e i=1/2 ln (-1)`

`Rightarrow log_e i=1/2 i pi quad` [ by Ex 4]

So ` ln i=frac{i pi}{2}`.

Thus the natural logarithm of `i` is `frac{i pi}{2}`.

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