Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 5 - P.S. Problem Solving - Page 420: 16


It is shown that: $\frac{\log_ax}{\log_{\frac{a}{b}}x}=1+\log_a\frac{1}{b}$

Work Step by Step

Use the Change-of-Base formula, the Quotient Property and the Power Property: $\frac{\log_ax}{\log_{\frac{a}{b}}x}=\frac{\frac{\log x}{\log a}}{\frac{\log x}{\log\frac{a}{b}}}=\frac{\log x}{\log a}·\frac{\log\frac{a}{b}}{\log x}=\frac{\log\frac{a}{b}}{\log a}=\frac{\log a-\log b}{\log a}=\frac{\log a}{\log a}-\frac{\log b}{\log a}=1+\frac{-\log b}{\log a}=1+\frac{\log b^{-1}}{\log a}=1+\frac{\log \frac{1}{b}}{\log a}=1+\log_a\frac{1}{b}$
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