 # Use the same-derivative argument, as was done to Alessandra Carrillo 2022-04-11 Answered
Use the same-derivative argument, as was done to prove the Productand Power Rules for logarithms, to prove the Quotient Ruleproperty.
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The objective is to prove the property $\mathrm{ln}\left(\frac{b}{x}\right)=\mathrm{ln}b-\mathrm{ln}x$
Differential $\mathrm{ln}\left(\frac{b}{x}\right)$

$=\frac{x}{b}\left(-\frac{b}{{x}^{2}}\right)$
$=\frac{-1}{x}$
Integrate both the sides of the equation with respect to x as,
$\mathrm{ln}\frac{b}{x}=\left(-\mathrm{ln}1\right)+c$
$c=\mathrm{ln}b$
Substitute $c=\mathrm{ln}b$ in the equation $\mathrm{ln}\left(\frac{b}{x}\right)=\left(-\mathrm{ln}x\right)+c$ as,
$\mathrm{ln}\frac{b}{x}=\left(-\mathrm{ln}x\right)+\mathrm{ln}b$
$\mathrm{ln}\left(\frac{b}{x}\right)=\mathrm{ln}b-\mathrm{ln}x$
Hence, Proved