Logarithmic Product identity

Formula

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

Introduction

A quantity can be split as a product of two or more quantities by the factorization. Likewise, the log of a quantity can be written as the logarithm of product of two or more quantities based on this concept.

$\implies$ $\log{(6)}$ $\,=\,$ $\log{(2 \times 3)}$

A special formula is required to find the logarithm of product of two or more quantities and it is called the product rule of logarithms. It is also called the logarithm of a product rule. Let’s learn what the product law of logarithms is, in mathematics.

According to the logarithmic product property, the logarithm of a product of quantities is equal to the summation of their logs.

Math form

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

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

Verification

Let’s learn how to verify the product law of the logarithms by an arithmetic example.

Take $m = 2$ and $n = 3$

$\log_{\displaystyle 10} 2 = 0.3010$ and $\log_{\displaystyle 10} 3 = 0.4771$. Now add both of them.

$\log_{\displaystyle 10} 2 + \log_{\displaystyle 10} 3 = 0.3010 + 0.4771$

$\log_{\displaystyle 10} 2 + \log_{\displaystyle 10} 3 = 0.7781$

Now calculate logarithm of product of both numbers

$\log_{\displaystyle 10} (2 \times 3) = \log_{\displaystyle 10} (6)$

$\implies \log_{\displaystyle 10} 6 = 0.7781$

$\therefore \,\,\,\,\, \log_{\displaystyle 10} (2 \times 3)$ $=$ $\log_{\displaystyle 10} 2 + \log_{\displaystyle 10} 3$ $=$ $0.7781$

Proof

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

Uses

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

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