The value of log 0 with base 10 is not defined. In this article, the concepts of how to find the value of log 0, using a common logarithmic function and natural logarithmic functions are explained.

What is the Value of Log 0?

To recall, in Mathematics, the logarithmic function defines an inverse function of exponentiation. The logarithmic function is defined by:

• if log_ab = x, then a^x = b

Where,

• x is the log of a number ‘b.’
• ‘a’ is the base of a logarithmic function.

Note: The variable “a” should be any positive integer, and a ≠1.

The logarithmic function is classified into two types. They are:

• Common Logarithmic Function – Logarithmic function with base 10
• Natural Logarithmic Function – Logarithmic function with base e

If the logarithmic function uses the base other than 10 or e, change it into either base 10 or base e by applying the change of base rule.

To eliminate the exponential functions, and to find the value of a variable, the log functions are functions.

How to Derive Log₁₀0 Value?

The log function of 0 to the base 10 is denoted by “log₁₀ 0”.

According to the definition of the logarithmic function,

Base, a = 10 and 10^x = b

We know that the real logarithmic function log_ab is only defined for b>0.

It is impossible to find the value of x, if a^x = 0,

i.e., 10^x = 0, where x does not exist.

So, the base 10 of logarithm of zero is not defined.

Therefore,

Log₁₀ 0 = Not Defined

Value of ln (0) or log_e 0

The natural log function of 0 is denoted by “log_e 0”. It is also known as the log function of 0 to the base e. The representation of the natural log of 0 is ln (0).

If, e^x =0, there is no number to satisfy the equation when x equals to any value.

Therefore, the value of log_e 0 is undefined

log_e 0 = ln (0) = Not defined

Log Values from 1 to 10

The logarithmic values from 1 to 10 to the base 10 are:

Ln Values from 1 to 10

The logarithmic values from 1 to 10 to the base e are:

Example Question from Log Values

Question: Find the value of y such that log_y 64 = 2

Solution:

Given that, log_y 64 = 2

According to the definition of the logarithm function,

if log_ab = x, then

a^x = b ….(1)

a = y, b= 64, x = 2

Substitute the values in (1), we get

y² = 64

Take square roots on both sides,

y = √8²

Therefore, the value of y is 8.

Visit BYJU’S- The Learning App to learn the values of natural log and common log, and also watch interactive videos to clarify the doubts.