Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 5 - Exponential and Logarithmic Functions - 5.5 Properties of Logarithms - 5.5 Assess Your Understanding - Page 306: 104


$\dfrac{f(x+h)-f(x)}{h}=\log_a {(\frac{x+h}{x})^{\frac{1}{h}}}$ Refer to the step-by-step part to see solution.

Work Step by Step

We know that $\log_a {x}-\log_a {y}=\log_a {(\frac{x}{y})}$. Hence, \begin{align*} \dfrac{f(x+h)-f(x)}{h}&=\dfrac{\log_a{(x+h)}-\log_a{x}}{h}\\ &=\frac{1}{h}(\log_a{(x+h)}-\log_a{x})\\ &=\frac{1}{h}\cdot \log_a {\left(\frac{x+h}{x}\right)} \end{align*} We know that $\log_a {x^n}=n\cdot \log_a {x}$. Hence, $\frac{1}{h}\cdot \log_a {\left(\frac{x+h}{x}\right)}= \log_a {\left(\frac{x+h}{x}\right)^{\frac{1}{h}}}$ Therefore, $\dfrac{f(x+h)-f(x)}{h}=\log_a {\left(\frac{x+h}{x}\right)^{\frac{1}{h}}}$
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