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: 35



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 $\log_{\sqrt 2} {4}=y$, then $\left(\sqrt 2\right)^y=4$ and we know that $4=2^2=((\sqrt 2)^2)^2=\left(\sqrt 2\right)^{4}.$ Thus, $\left(\sqrt 2\right)^{y}=\left(\sqrt 2\right)^{4}$. 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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