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:

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.

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.