# Common Logarithm and Natural Logarithm: Definition, Examples, Difference | Log vs Ln

From the introduction to logarithm, we know that the value of a logarithm does not make any sense without the base. In the topic of logarithms, we often hear the terms common logarithm and natural logarithm. In this section, we will discuss them.

## Common Logarithm

The logarithm of a number with base 10 is called the common logarithm of that number. It is also known as decimal logarithm.

For example, the logarithm of 7 with base 10, that is, log10 7 is called the common logarithm of 10.

Note: The common logarithm of a number M is usually denoted as $\log M.$ So both log10M and $\log M$ have the same meaning. In other words, both represent the same number.

## Natural Logarithm

The logarithm of a number with base $e$ is called the natural logarithm of that number. It is also known as Napierian logarithm, named after John Napier – a Scottish Mathematician. Here the number $e$ is an irrational number whose value is equal to the following infinite sum:
$e=\sum_{n=0}^\infty \frac{1}{n!}$

Note: The natural logarithm of a number $x$ is usually denoted as $\ln x.$ From the above discussion, we see that the numbers $\log x$ and $\ln x$ are different.

## Log vs Ln

The difference between log and ln is provided in the table below.

Solved Examples of common logarithms:

Solution:

We need to find log10 10

As logaa=1, we have log1010=1.

So the common logarithm of 10 is 1.

Solution:

Note that 1000=103.

So log10 1000 = log10103 = 3

$[\because \log_a a^n=n]$

So the common logarithm of 1000 is 3.

Solved Examples of natural logarithms:

Solution:

We need to find logee

As logaa =1, we have loge e=1.

So the natural logarithm of e is 1.

Solution:

As the natural logarithm has base e, we have to find loge (-1)

It is known that -1 = e

So loge (-1) = loge e = iπ

$[\because \log_a a^n=n]$

So the natural logarithm of -1 is iπ.

Solution:

We need to find loge i

We known that i = $e^{i \frac{\pi}{2}}$

So loge i = loge $e^{i \frac{\pi}{2}}$ = iπ/2.

$[\because \log_a a^n=n]$

So the natural logarithm of i is iπ/2.