Math Doubts

Logarithm of a Product identity

Formula

$\log_{b}{(xy)}$ $\,=\,$ $\log_{b}{x}$ $+$ $\log_{b}{y}$

What is Logarithm of a Product Law?

The logarithm of a product of two or more quantities is mathematically equal to the sum of their logarithms. It is a natural property of the logarithms and it is called the logarithm of a product rule.

Let’s understand what the product rule of logarithms practically is from an arithmetic example.

Example

$\log{6}$ $\,=\,$ $0.7781$

The common logarithm of $6$ is a decimal number and its approximate value is $0.7781$. The number $6$ can be factored as a product of $2$ and $3$.

$\therefore\,\,\,$ $\log{(2 \times 3)}$ $\,=\,$ $0.7781$

Therefore, the logarithm of product of two numbers $2$ and $3$ is equal to $0.7781$. Now, let’s find the values of logarithms of numbers $2$ and $3$ from log table.

  1. $\log{2}$ $\,=\,$ $0.3010$
  2. $\log{3}$ $\,=\,$ $0.4771$

Now, add the above two quantities to understand the logarithm of a product property practically.

$\implies$ $\log{2}$ $+$ $\log{3}$ $\,=\,$ $0.3010$ $+$ $0.4771$

$\implies$ $\log{2}$ $+$ $\log{3}$ $\,=\,$ $0.7781$

Actually, the common logarithm of $6$ is equal to $0.7781$.

$\implies$ $\log{2}$ $+$ $\log{3}$ $\,=\,$ $\log{6}$

$\implies$ $\log{2}$ $+$ $\log{3}$ $\,=\,$ $\log{(2 \times 3)}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\log{(2 \times 3)}$ $\,=\,$ $\log{2}$ $+$ $\log{3}$

The above simple arithmetic example explains us that the logarithm of a product is equal to the summation of their logs. This logarithmic product rule can be used as a formula in mathematics.

Log Product Identity

Let $m$ and $n$ denote two literal quantities, and $b$ represents another quantity in algebraic form. The product of the quantities $m$ and $n$ is written as either $m \times n$ or $m.n$ in mathematics, and the log of their product to the base $b$ is written in the following mathematical forms.

$(1).\,\,$ $\log_{b}{(m \times n)}$

$(2).\,\,$ $\log_{b}{(m.n)}$

The logarithms of both literal quantities $m$ and $n$ to the base $b$ are written mathematically as follows.

$(1).\,\,$ $\log_{b}{(m)}$

$(2).\,\,$ $\log_{b}{(n)}$

According to the logarithms, the logarithm of a product is equal to the sum of their logarithms. Therefore, the mathematical relation between them is expressed in mathematics as follows.

$\log_{b}{(m.n)}$ $\,=\,$ $\log_{b}{(m)}+\log_{b}{(n)}$

This mathematical property is called the product identity of the logarithms.

Extension

$\log_{b}{(m \times n \times o \times p \cdots)}$ $\,=\,$ $\log_{b}{m}$ $+$ $\log_{b}{n}$ $+$ $\log_{b}{o}$ $+$ $\log_{b}{p}$ $+$ $\cdots$

Proof

Learn how to derive the logarithm of product of two or more quantities is equal to the sum of their logs.

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Uses

Learn how to use property of product law of logarithms in mathematics.

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