$\log_{b}{(m \times n)}$ $\,=\,$ $\log_{b}{m}+\log_{b}{n}$
The logarithm of product of quantities equals to sum of their logs is called the product rule of logarithms.
Learn how to prove logarithm of product of two or more quantities is equal to sum of their logs.
Learn how to use property of product law of logarithms in mathematics.
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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