Value of Log 0

Log of Zero

In Mathematics, most of the researchers used logarithms to transform multiplication and division problems into addition and subtraction problems before the process of calculus has been found out. Logarithms are continuously used in Mathematics and Science as these both subjects contend with large numbers. In this article, we will discuss the log 0 value (log 0 is equal to not defined) and method to derive the log 0 value through common logarithm functions and natural logarithm functions.

Logarithm Functions

Before deriving Log 0 value, let us discuss logarithm functions and their classifications. A logarithm function is an inverse function to an exponential. Mathematically logarithm function is defined as:

If Logab = x, then ax =b

Where, a

  • “x” is considered as the log of a number 

  • “a” is considered as the base of a logarithm function.

Note= The variable “a” should always be a positive integer and not equals to 1.

Classification of Logarithm Function

Common logarithm functions – Common logarithm function is the logarithm function with base 10 and is denoted by log10 or log.

F(x) = log10 x

Natural logarithm functions - Natural logarithm functions is the logarithm functions with base e and is denoted by loge

F(x) = loge x

Log functions are used to find the value of a variable and eliminate the exponential functions. Tabular data will be updated soon.

What is the Value of Log 0 and How it can be Derived?

Here, we will discuss the procedure to derive the Log 0 value.

The log functions of 0 to the base 10 is expressed as Log10 0

On the basis of the logarithm function,

Base = 10 and 10x = b

As we know,

The logarithm function logab can only be defined if b > 0, and it is quite impossible to find the value of x if ax = 0.

Therefore, log0 10 or log of 0 is not defined.

The natural log function of 0 is expressed as loge 0. It is also known as log function 0 to the base e. The representation of natural log of 0 is Ln

If ex = 0

No number can agree with the equation when x equals to any value.

Hence, log 0 is equal to not defined.

Loge 0 = In (0) = Not defined

Logarithm Value Table from 1 to 10

Logarithm Values to the Base 10 are:

Log 1


Log 2


Log 3


Log 4


Log 5


Log 6


Log 7


Log 8


Log 9


Log 10



Ln Values table from 1 to 10

Logarithm Values to the Base e are:

In (1)


In (2)


In (3)


In (4)


In (5)


In (6)


In (7)


In (8)


In (9)


In (10)



Solved Example

1. Solve for y in log₂ y =6

Solution: The logarithm function of the above function can be written as 26 = y

                  Hence, 25 =2 x 2 x 2 x 2 x 2 x 2 =64 or Y =64

2. Find the value of x such that log 81 =2


Given that, log 81=2

On the basis of Logarithm definition

If logx b=x

a= b – (1)

a=x, b= 81, x =2

Substituting the value in equation (1), we get

 x2 =81

Taking square root on both sides we get,

x = 9

Therefore, the value of x = 9

Fun Facts

  • The logarithm with base 10 is known as common or Briggsian, logarithms and can be written as log n. They are usually written as without base.

  • Concept of Logarithm was introduced by John Napier in the 17th century

  •   The logarithm is the inverse process of exponentiation.

  •  The first man to use Logarithm in modern times was the German Mathematician, Michael Stifel (around 1487 -1567).

  •  According to Napier, logarithms express ratios.

  •  Henry Briggs proposed to make use of 10 as a base for logarithms.

Quiz Time

1. Which of the following is incorrect?

a. Log10 = 1

b. Log( 2+3) = Log( 2x3)

c. Log10 1 = 0

d. Log ( 1+2+3) = log 1 + log 2+ log 3

2. If log a/b + log b/a = log( a+b), then:

a. a + b=1

b. a – b = 1

c. a = b

d. a² - b² = 1

FAQ (Frequently Asked Questions)

1. What is Logarithm?

A. The logarithm is defined as the exponent or power to which a base must be raised to get some new number. It is a convenient approach to express large numbers.. Through logarithm, the multiplication of large numbers can be easily resolved speedily. Some common properties of logarithm which proved multiplication and division of logarithms can even be written in the form of logarithm of addition or subtraction. 

Mathematically Logarithm can be expressed as:

X is the logarithm of n to the base b, if bx= n, it can be written as x = logn

For Example - 23 = 8, in which 3 is the logarithm of 8 to base 2 or can be written as 3= log28.

Similarly 102 = 100 can be written as 2 =log10100

The logarithm with base 10 is commonly known as common or Briggsian logarithms.

2. Explain the Properties of Logarithm.

A. The four basic properties of logarithm which helps to resolve problems based on logarithm are:

1. Logb (mn) = Logb m + Logb n

This property says that the multiplication of two logarithm values is equivalent to the addition of the individual logarithm.

For example – log2 (4y) = log3(2) + log3(y)

2. Logb (m/n) = Logb m - Logb n 

This property says that the division of two logarithm values is equivalent to the subtraction of the individual logarithm.

For example – log₂ (4y) = log₃ (2) + log₃(y)

3. Logb (m n) = n logbm

Above property is the exponential rule of the logarithm. The logarithm of m along with the rational exponent is equivalent to the exponent times its logarithm.

4. Logb m = loga m / loga 

When two numbers are divided with the same base, then the exponents will be subtracted.