Common Logarithm and Natural Logarithm: Definition, Examples, Difference

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, 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.

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 log₁₀ 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 log₁₀ 10

As log_aa=1, we have log₁₀10=1.

So the common logarithm of 10 is 1.

Problem 2: Calculate the common logarithm of 1000.

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:

Problem 3: What is the natural logarithm of e.

Solution:

We need to find log_ee

As log_aa =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.

FAQs

Q1: What is Log in math?

Answer: Log denotes the logarithm with base 10. For example, Log of x is equal to log₁₀x.

Q2: What is Ln in math?

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

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.

Tathagata

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.

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