Value of Log 0

Value of log 0 is undefined. It is not a real number as we can never get zero by raising any power of any number. This can never be reached to zero but can be approached using negative power or infinitely large value. In this article, we will discuss the value of Log 0 along with basic understanding of logarithms. We will also discuss how to derive Log 0 both Log₁₀ 0 and Log_e 0.

What is the Value of Log 0?

The value of the Log 0 is not defined for any base of the logarithm. In other words, we can say that the value of Log 0 is infinity. We cannot get any power which when raised to any number gives the result 0 so, the Log of 0 is undefined. There is no value of b and x that satisfies the expression b^x = 0 where b is the base of Log and x is the value of log function

Value of Log 0 = Undefined

or

Value of Log 0 = ∞

Note: Value of Log 0 in any base is undefined.

What is Logarithm?

A logarithm is the inverse operation of the exponential function. The logarithm functions are denoted as f(x) = log_b z where b is the base of the logarithm and z is the number. The formula for converting the logarithm to exponent is given as:

x^a = q ⇔ a = log_x q

Types of Logarithms

The two types of logarithms are:

• Common Logarithm (log₁₀ x): The logarithm with base 10 is called as common logarithm.

• Natural Logarithm (log_e18): The logarithm with base e is called as natural logarithm.

How to Derive Value of Log 0

To derive the value of Log 0 we use the fundamental formula of conversion of logarithm to exponent or vice versa i.e., x^a = q ⇔ a = log_x q. The value of Log_e 0 and Log₁₀ 0 both are not defined as we cannot have any number whose any power is equivalent to 0.

Derivation of Value of Log_e 0

To derive the value of Log_e 0 i.e., not defined we will use the log to exponent conversion formulae.

Let y = Log_e 0

We know that,

x^a = q ⇔ a = log_x q

By the above formula

y = Log_e 0 ⇔ e^y = 0

We cannot find any value of y which satisfies e^y = 0. So, the value of Log_e 0 is undefined.

Value of Log_e 0 = Undefined

or

Value of Log_e 0 = ∞

Derivation of Value of Log₁₀ 0

To derive the value of Log_e 0 i.e., not defined we will use the log to exponent conversion formulae.

Let y = Log_e 0

We know that,

x^a = q ⇔ a = log_x q

By the above formula

y = Log₁₀ 0 ⇔ 10^y = 0

We cannot find any value of y which satisfies 10^y = 0. So, the value of Log_e 0 is undefined.

Value of Log₁₀ 0 = Undefined

or

Value of Log₁₀ 0 = ∞

Some other Log Values

Some other values of logarithm with base e and 10 are listed in the following table:

Conclusion

In conclusion, value of log 0 in any base is not defined either it is common or natural. Logarithms are an important concept in math because they provide a powerful tool for simplifying calculations involving exponential growth, expressing relationships between quantities with different scales, and solving equations involving exponential functions.