Quotient Rule of Logarithms
Formula
$\log_{b}{\bigg(\dfrac{x}{y}\bigg)}$ $\,=\,$ $\log_{b}{x}$ $-$ $\log_{b}{y}$
What is the quotient rule of logarithms?
The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator.
This is the fundamental definition of the quotient law of logarithms, one of the most important laws of logarithms. In mathematics, the quotient law of logarithms is used as a formula to convert a quotient into subtraction, and it is widely applied to simplify logarithmic expressions.
In arithmetic, the quotient of two numbers is obtained by dividing the numerator by the denominator. However, the logarithm of a quotient cannot be evaluated in the same way.
Example
In arithmetic, $\dfrac{6}{2} \,=\, 3$
But, $\log{\bigg(\dfrac{6}{2}\bigg)}$ $\,\ne\,$ $\dfrac{\log{(6)}}{\log{(2)}}$
To find the logarithm of quotient, we use a special logarithmic identity called the quotient law of logarithms.
Examples of Quotient rule of logarithms
Now, let’s know the logarithm quotient rule through a simple and clear example.
Example
Evaluate $\log_{2}{\bigg(\dfrac{16}{2}\bigg)}$
First, divide the number $16$ by $2$ to obtain the quotient.
$=\,\,$ $\log_{2}{\bigg(\dfrac{\cancel{16}}{\cancel{2}}\bigg)}$
$=\,\,$ $\log_{2}{(8)}$
As per the definition of the logarithm, the logarithm of $8$ to base $2$ is $3$.
$\,\,\,\,\therefore\,\,\,\,$ $\log_{2}{\bigg(\dfrac{16}{2}\bigg)}$ $\,=\,$ $3$
Now, let’s find the logarithms of the numerator $16$ and denominator $2$ to base $2$.
- $\log_{2}{(16)}$ $\,=\,$ $4$
- $\log_{2}{(2)}$ $\,=\,$ $1$
Finally, subtract the logarithm of denominator from the logarithm of numerator to find their difference mathematically.
$\implies$ $\log_{2}{(16)}$ $-$ $\log_{2}{(2)}$ $\,=\,$ $4$ $-$ $1$
$\,\,\,\,\therefore\,\,\,\,$ $\log_{2}{(16)}$ $-$ $\log_{2}{(2)}$ $\,=\,$ $3$
Now, let’s compare both results to know how the quotient rule for logarithms works.
$\,\,\,\,\therefore\,\,\,\,$ $\log_{2}{\bigg(\dfrac{16}{2}\bigg)}$ $\,=\,$ $\log_{2}{(16)}$ $-$ $\log_{2}{(2)}$ $\,=\,$ $3$
The above numerical example helps you to understand why the logarithm of a quotient is equal to the difference between the logarithms of its numerator and denominator. This logarithmic property is known as the quotient rule of logarithms. The following examples will help you to understand and apply this rule more clearly.
- $\log_{3}{\bigg(\dfrac{27}{9}\bigg)}$ $\,=\,$ $\log_{3}{(27)}$ $-$ $\log_{3}{(9)}$
- $\log{\bigg(\dfrac{3a}{4b}\bigg)}$ $\,=\,$ $\log{(3a)}$ $-$ $\log{(4b)}$
- $\log_{7}{\bigg(\dfrac{25}{3}\bigg)}$ $\,=\,$ $\log_{7}{(25)}$ $-$ $\log_{7}{(3)}$
How to Write the Log Quotient Rule
The quotient property of logarithms is expressed mathematically as follows.
$\log_{b}{\bigg(\dfrac{x}{y}\bigg)}$ $\,=\,$ $\log_{b}{(x)}$ $-$ $\log_{b}{(y)}$
Domain Conditions
It is one of the most commonly used logarithmic identities in mathematics; however, it is important to know the conditions under which the logarithmic quotient rule applies.
- $b$ denotes a quantity ($b > 0$ and $b \ne 1$) and the base of logarithm.
- $x$ and $y$ represent two quantities ($x > 0$ and $y > 0$).
- The quotient rule is valid only when all arguments of the logarithm are positive.
This explains how the log of a quotient expands into the subtraction of the logs of numerator and denominator.
Derivation of the Log Quotient Law
Let’s learn how to prove that the logarithm of a quotient is equal to the difference between the logarithms of numerator and denominator.
You have learned what the quotient law in logarithms is, understood 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.
$\log_{b}{(xy)}$ $\,=\,$ $\log_{b}{(x)}$ $+$ $\log_{b}{(y)}$
$\log_{b}{\big(x^n\big)}$ $\,=\,$ $n \times \log_{b}{(x)}$
$\log_{b}{(x)}$ $\,=\,$ $\dfrac{\log_{c}{(x)}}{\log_{c}{(b)}}$