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 Loge 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 bx = 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) = logb 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:
xa = q ⇔ a = logx 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 (loge18): 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., xa = q ⇔ a = logx q. The value of Loge 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 Loge 0
To derive the value of Loge 0 i.e., not defined we will use the log to exponent conversion formulae.
Let y = Loge 0
We know that,
xa = q ⇔ a = logx q
By the above formula
y = Loge 0 ⇔ ey = 0
We cannot find any value of y which satisfies ey = 0. So, the value of Loge 0 is undefined.
Value of Loge 0 = Undefined
or
Value of Loge 0 = ∞
Derivation of Value of Log10 0
To derive the value of Loge 0 i.e., not defined we will use the log to exponent conversion formulae.
Let y = Loge 0
We know that,
xa = q ⇔ a = logx q
By the above formula
y = Log10 0 ⇔ 10y = 0
We cannot find any value of y which satisfies 10y = 0. So, the value of Loge 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:
| Number (x) | ln (x) or loge x | log (x) or log10 x |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 0.693147 | 0.30103 |
| e | 1 | 0.43429 |
| 3 | 1.098612 | 0.47712 |
| 4 | 1.386294 | 0.60206 |
| 5 | 1.609438 | 0.69897 |
| 6 | 1.791759 | 0.77815 |
| 7 | 1.94591 | 0.84510 |
| 8 | 2.079442 | 0.90309 |
| 9 | 2.197225 | 0.95424 |
| 10 | 2.302585 | 1.00000 |
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.
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FAQs on Value of Log 0
What is the value of log 0?
The value of log 0 is undefined in both common logarithm (base 10) and natural logarithm (base e).
Why is log 0 undefined?
Logarithms represent the exponent to which a base must be raised to produce a given number. However, there is no real number that, when raised to any power, results in zero. Therefore, log 0 does not have a real value.
Can log 0 have a defined value in certain contexts?
In some specialized mathematical contexts, such as complex analysis, log 0 may be assigned a limit value.
Is the value of log 0 is infinity?
No, we can not say that the value of log 0 is infinity as infinity is not the number it only represents the undefined nature of value of log 0.