# 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 $n = 3$

$\log_10} 2 = 0.301$ and $\log_10} 3 = 0.477$. Now add both of them.

$\log_10} 2 + \log_10} 3 = 0.3010 + 0.477$

$\log_10} 2 + \log_10} 3 = 0.778$

Now calculate logarithm of product of both numbers

$\log_10} (2 \times 3) = \log_10} (6$

$\implies \log_10} 6 = 0.778$

$\therefore \,\,\,\,\, \log_10} (2 \times 3$ $=$ $\log_10} 2 + \log_10}$ $=$ $0.7781$

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