Logarithm of One identity
Formula
$\log_{b}{(1)} \,=\, 0$
What is Log of 1 rule?
The number $1$ is the first natural number. So, it cannot be split on the basis of any other number. Mathematically, it can be expressed that the log of one to any base number is zero, and it is called “logarithm of one” property.
If you are a beginner, the log of $1$ law may confuse you. So, let’s understand from a simple understandable example.
Example
Can you split $6$ on the basis of $2$?
It is possible. The number $6$ can be written in product form on the basis of number $2$ as follows.
$6$ $\,=\,$ $2 \times 2 \times 2$
Now, count how many $2$s are there in the product, and they are three.
$\implies$ $6$ $\,=\,$ $\underbrace{2 \times 2 \times 2}_{\displaystyle 3}$
So, it is called the logarithm of $6$ to base $2$ is $3$. It is mathematically written as follows.
$\,\,\,\, \therefore \,\,\,\,\,$ $\log_{2}{6}$ $\,=\,$ $3$
Can you split $2$ on the basis of $6$?
It is not possible because the number $2$ is smaller than $6$.
Look at the following examples.
$(1).\,\,$ $\log_{5}{(1)} \,=\, 0$
$(2).\,\,$ $\log_{12}{(1)} \,=\, 0$
$(3).\,\,$ $\log_{617}{(1)} \,=\, 0$
You can understand that the value of the logarithm of one to any base number is zero. Hence, the base of the logarithm is denoted by an algebraic literal $b$ and the logarithm of one law can be expressed in algebraic form as follows.
$\therefore \,\,\,\,$ $\log_{b}{(1)} \,=\, 0$
Proof
Learn how to prove the logarithm of one property in algebraic form for using it as a formula in mathematics.