# Common Logarithms and Natural Logarithms

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, $\log_{10} 7$ is called the common logarithm of 10.

Note: The common logarithm of a number $M$ is usually denoted as $\log M.$ So both $\log_{10} M$ and $\log M$ have the same meaning. In other words, both represent the same number.

Solved Examples of common logarithms:

Problem 1: Find the common logarithm of $10$

Solution: We need to find $\log_{10} 10$

As $\log_a a=1,$ we have $\log_{10} 10=1$

So the common logarithm of $10$ is $1.$

Problem 2: Calculate the common logarithm of $1000$

Solution: Note that $1000=10^3$

So $\log_{10} 1000=\log_{10}10^3=3$

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

So the common logarithm of $1000$ is $3.$

#### 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.

Solved Examples of natural logarithms:

Problem 3: What is the natural logarithm of $e$

Solution: We need to find $\log_e e$

As $\log_a a=1,$ we have $\log_e e=1$

So the natural logarithm of $e$ is $1.$

Problem 4: (Natural logarithm of a negative number)

Find the natural logarithm of $-1$

Solution: As the natural logarithm has base $e$, we have to find $\log_e -1$

It is known that $-1=e^{i \pi}$

So $\log_e (-1)=\log_e e^{i \pi}=i \pi$

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

So the natural logarithm of $-1$ is $i \pi.$

Problem 5: (Natural logarithm of an imaginary number)

Find the natural logarithm of $i$

Solution: We need to find $\log_e i$

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

So $\log_e (-1)=\log_e e^{i \frac{\pi}{2}}=\frac{i\pi}{2}$

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

So the natural logarithm of $-1$ is $\frac{i\pi}{2}.$

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