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 - 5.4 Logarithmic Functions - 5.4 Assess Your Understanding - Page 295: 31



Work Step by Step

The definition of the logarithmic function says that $y=\log_a{x}$ if and only if $a^y=x$. Also, $a\gt0,a\ne1$ and $x\gt0$. Hence $\log_{\frac{1}{2}} {16}=y$, then $\left(\frac{1}{2}\right)^y=16$ and we know that $16=\left(\frac{1}{2}\right)^{-4}.$ Thus, $\left(\frac{1}{2}\right)^{y}=\left(\frac{1}{2}\right)^{-4}$. We know that $a^b=a^c\longrightarrow b=c$ if $a\ne1,a\ne-1$ (which applies here), hence $y=-4$.
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