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.

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**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:**

**Prob****lem**** 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 $1.$

**Proble****m**** 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**

**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:

**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:**

**Probl****em**** 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 $1.$

**Proble****m**** 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 $i \pi.$

**Probl****em**** 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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