Identity Property of Logarithms
Formula
$\log_{b}{(b)} \,=\, 1$
What is the Identity rule of Logarithms?
The logarithm of a quantity that is the same as its base is always equal to one.
This is the definition of the identity property of logarithms, one of the basic laws of logarithms. It is used in mathematics to evaluate the logarithm of a quantity that is equal to its base and is widely used in simplifying logarithms.
In general English, the word ‘identity’ refers to sameness. In this case, the argument of the logarithm is the same as the base of the logarithm; therefore, this logarithmic property is called the identity law of logarithms.
Examples of Identity rule of logarithms
Now, let us understand the logarithm identity property using a numerical example.
Example
Evaluate $\log_{3}{(3)}$
- The argument of the logarithm is $3$.
- The base of the logarithm is also $3$.
In this example, the argument and the base of the logarithm are the same. The argument is written once in the form of the base of the logarithm. Therefore, the logarithm of $3$ to base $3$ is equal to $1$, according to the fundamental definition of logarithms.
$\therefore\,\,\,$ $\log_{3}{(3)}$ $\,=\,$ $1$
The above numerical example shows why the logarithm of a number that is the same as its base is equal to one. This logarithmic property is called the identity rule of logarithms. The following examples will help you understand this log rule more easily.
- $\log_{2}{(2)} \,=\, 1$
- $\log_{e}{(e)} \,=\, 1$
- $\log_{10}{(10)} \,=\, 1$
How to Write the Log Identity Rule
The identity property of logarithms is expressed mathematically as follows.
$\log_{b}{(b)} \,=\, 1$
Domain Conditions
It is one of the most commonly used basic logarithmic identities in mathematics. However, students must clearly understand the conditions under which the identity rule of logarithms can be applied.
- $b$ denotes a quantity and the argument of the logarithms ($b > 0$)
- $b$ also represents the base of the logarithms ($b \ne 1$).
This explains how the log of a quantity that is the same as its base is always equal to one.
Derivation of the Log Identity Law
Learn how to prove that the logarithm of a number that is the same as its base is always equal to one.
You have learned what the identity law of logarithms is, studied its proof, and practiced solving examples to understand how this logarithmic formula works in mathematics.
Properties of Logarithms
Learn similar logarithmic properties that make it easier to simplify expressions and solve problems.
$\log_{b}{(1)} \,=\, 0$
$b^{\displaystyle \log_{b}{(x)}} \,=\, x$