The Value of log 0 with base 10 and e
How do you determine the value of Log 0?
Before determining the value of Log 0 let’s understand the basics of a logarithmic function.
What is a Logarithmic Function?
In mathematics, Logarithmic Function is known to be the inverse function of an exponential function.
The logarithmic function is usually defined as:
\log _{a} b=x
The equivalent exponential form of the log function is:
a^{x}=bHere, x is the log of a number b, and a is the base of the logarithmic function.
And, a is a positive integer and a ≠1.
The logarithmic function has two types, which are:
(1) Common Logarithmic Function
(2) Natural Logarithmic Function
Common Logarithmic Function
The function which uses ‘10’ as the base is a common logarithmic function.
\log _{a} b=x \Rightarrow a^{x}=b
\log _{10} b=x \Rightarrow 10^{x}=b, \text { [Putting base, } \mathrm{a}=10 \text { ] }
Natural Logarithmic Function
In this log function, the base used is ‘e’.
\log _{a} b=x \Rightarrow a^{x}=b
\log _{e} b=x \Rightarrow e^{x}=b, \text { [Putting base, } \mathrm{a}=\mathrm{e} \text { ] }
Natural Logarithm is usually represented as ‘Ln’.
\log _{10} 0
As per the definition of logarithmic function:
\log _{a} b=x \Rightarrow a^{x}=b
\text{Put } \mathrm{a}=10 \text{ and }\mathrm{b}=0
\log _{10} 0=x \Rightarrow 10^{x}=0
We know that the real log function is defined only for b > 0.
Therefore, it is impossible to determine the value of x for which 10^x=0.
Hence, log 0 to the base 10 is undefined.
\log _{e} 0 \text { or } \ln 0
As per the definition of logarithmic function:
\log _{a} b=x \Rightarrow a^{x}=b
\text{Put } \mathrm{a}=e \text{ and } \mathrm{b}=0
\log _{e} 0=x \Rightarrow e^{x}=0
We know that the real log function is defined only for b > 0.
Therefore, it is impossible to determine the value of x for which e^x=0.
Hence the value of \log_{e}0 \text{ or }\ln 0 is undefined.
Note: Here ‘e’ is an exponential constant and its value is 2.7182818 (rounded to 7 digits). It is an important mathematical constant used to ease out exponential calculations.
Log Table
The table below gives the value of the common and natural logarithm of numbers from 1 to 10.
Frequently Asked Questions
Q1. What do you mean by logarithmic function?
Ans: Logarithmic Function is known to be the inverse function of an exponential function.
The logarithmic function is usually defined as:
\log _{a} b=x
The equivalent exponential form of the log function is:
a^{x}=b
Here, x is the log of a number b, and a is the base of the logarithmic function.
And, a is a positive integer and a ≠1.
Q2. What are the Common and Natural Logarithmic functions?
Ans: The logarithmic function to the base ‘10’ is known as the Common Logarithmic function. Whereas when the base is ‘e’ it is a Natural Logarithmic function.
Q3. What are the common and natural log values of zero?
Ans: The value of both common and natural log of zero are both undefined.