Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 7 - Functions - Exercise Set 7.2 - Page 415: 33

Answer

$\log_{b}\frac{x}{y}={\log_{b}x}-{\log_{b}y}\\ proof:\\ let\,\,log_{b}\frac{x}{y} =f \,\,\,,\log_{b}x=g\,\,\,,\log_{b}y=h \\ log_{b}\frac{x}{y} =f \Leftrightarrow b^f=\frac{x}{y} \\ \log_{b}x=g \Leftrightarrow b^g=x \\ \log_{b}y=h \Leftrightarrow b^h=y \\ so\,\,b^f=\frac{x}{y}=\frac{b^g}{b^h}=b^{g-h}\\ \therefore f=g-h \\ \Rightarrow \log_{b}\frac{x}{y}={\log_{b}x}-{\log_{b}y}\\ $

Work Step by Step

$\log_{b}\frac{x}{y}={\log_{b}x}-{\log_{b}y}\\ proof:\\ let\,\,log_{b}\frac{x}{y} =f \,\,\,,\log_{b}x=g\,\,\,,\log_{b}y=h \\ log_{b}\frac{x}{y} =f \Leftrightarrow b^f=\frac{x}{y} \\ \log_{b}x=g \Leftrightarrow b^g=x \\ \log_{b}y=h \Leftrightarrow b^h=y \\ so\,\,b^f=\frac{x}{y}=\frac{b^g}{b^h}=b^{g-h}\\ \therefore f=g-h \\ \Rightarrow \log_{b}\frac{x}{y}={\log_{b}x}-{\log_{b}y}\\ $
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