Question

Prove the law of logarithm with example ​

Answer 1

Step-by-step explanation:

$\rm \: Let \: \: log_{a}(m) = \alpha \: \: and \: \: log_{a}(n) = \beta$

$\rm \: \implies \: m = {a}^{ \alpha } \: \: and \: \: n = {a}^{ \beta }$

Now,

$\rm \: log_{a}(mn) = log_{a}( {a}^{ \alpha }. {a}^{ \beta } )$

$\rm \implies \: log_{a}(mn) = log_{a}( {a}^{ \alpha + \beta } )$

$\rm \implies \: log_{a}(mn) = (\alpha + \beta )log_{a}(a )$

$\rm \implies \: log_{a}(mn) = (\alpha + \beta ).1$

$\rm \implies \: log_{a}(mn) = \alpha + \beta$

$\rm \implies \: log_{a}(mn) = log_{a}(m) + log_{a}(n)$