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$\log_{a}\left(\frac{x}{y}\right) = \log_{a}(x) - \log_{a}(y)$
in Grade 12 Maths by Platinum (110k points) | 51 views

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Second law of logarithms states that $\log{(\dfrac{a}{b})} = \log{a} - \log{b}$

Let the base of the logs be a everywhere.
let $\log{x}$ be $b$ and $\log{y}$ be $c$
this implies that $a^b=x$ and $a^c=y$

now $\dfrac{x}{y}= \dfrac{a^b}{a^c}$
and $\dfrac{a^b}{a^c}=\dfrac{a^b}{a^-c}=a^b-c$

This implies that $\log{(\dfrac{x}{y})}=b-c$
$b=\log{x}$ and $c=\log{y}$
therefore $\log{(\dfrac{x}{y})} =\log{x}-\log{y}$  , proven.
by Diamond (80.1k points)
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