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.

Table of Contents

**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.

**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.

**Log vs Ln**

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

Log | Ln |

It represents logarithms with base 10. | It represents logarithms with base e. |

Known as common logarithm. | Known as natural logarithm. |

Log of x is written as log_{10} x | Ln of x is written as log_{e} x |

Exponential Form: 10^{x}=y | Exponential Form: e^{x}=y |

**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.

**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^{iπ}

So log_{e} (-1) = log_{e} e^{iπ} = iπ

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

So the natural logarithm of -1 is iπ.

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

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

So the natural logarithm of i is iπ/2.

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**FAQs**

**Q1: What is Log in math?**

Answer: Log denotes the logarithm with base 10. For example, Log of x is equal to log_{10}x.

**Q2: What is Ln in math?**

Answer: Ln denotes the logarithm with base e. For example, Ln of x is equal to log_{e}x.

**Q3:**

**What is the difference between ln and log?**Answer: Log is defined for base 10 whereas ln is defined for base e. This is the difference between log and ln.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.