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: 41


$x \ne 0$.

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 $x\gt0$. By the definition of the logarithmic function we can only take the logarithm of a positive number, hence in the expression $\log_a {A}$ or $\ln {A}$, $A$ must be positive. Hence $x^2\gt 0$, but because $x^2$ is always non-negative, the only value that is not good is when $x^2=0$ and this is when $x=0$, hence the domain is all numbers apart from $x=0$: $x \ne 0$.
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