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.