College Algebra (6th Edition)

Published by Pearson
ISBN 10: 0-32178-228-3
ISBN 13: 978-0-32178-228-1

Chapter 4 - Exponential and Logarithmic Functions - Exercise Set 4.3: 33

Answer

$$\log_b\Bigg(\frac{x^2y}{z^2}\Bigg)=2\log_bx+\log_by-2\log_bz$$

Work Step by Step

$$A=\log_b\Bigg(\frac{x^2y}{z^2}\Bigg)$$ First, we apply the Quotient Rule, which states $$\log_b\frac{M}{N}=\log_b M-\log_bN$$ ($M, N, b\in R, M\gt0, N\gt0, b\gt0, b\ne1$) That means, $$A=\log_b(x^2y)-\log_b(z^2)$$ Now, for $\log_b(x^2y)$, we apply the Product Rule, which states $$\log_bMN=\log_bM+\log_bN$$ ($M, N, b\in R, M\gt0, N\gt0, b\gt0, b\ne1$) Therefore, $$A=\log_b(x^2)+\log_b y-\log_b(z^2)$$ Finally, for $\log_b(x^2)$ and $\log_b(z^2)$, Power Rule can be applied, $$\log_bM^p=p\log_bM$$ ($M, b, p\in R, M\gt0, b\gt0, b\ne1$) So, $$A=2\log_bx+\log_by-2\log_bz$$
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