# Product Law of Logarithms

## Formula

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

The logarithm of product of quantities equals to sum of their logs is called the product rule of logarithms.

### Proof

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

### Use

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

#### Verification

The fundamental product rule of logarithm can be verified in mathematics by using numerical method.

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

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

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

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

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