Intermediate Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-894-7
ISBN 13: 978-0-13417-894-3

Chapter 9 - Section 9.3 - Logarithmic Functions - Exercise Set: 51

Answer

$(-\infty, 2) \cup (2, +\infty)$

Work Step by Step

RECALL: The domain of the logarithmic function $f(x) = \log_a{x}$ is $x \gt 0$. Thus, the domain of the given function is the set of all real numbers such that: $(x-2)^2 \gt 0$. Note that the value of the square any number will always be positive except when the number is zero. Thus, The value of $(x-2)^2$ will only be 0 when $(x-2)$ itself is zero. The value of $x-2$ is equal to zero only when $x=2$ This means that $(x-2)^2 \gt 0$ for all real numbers except 2. Therefore, the domain of the given function is the set of all real numbers except 2. In interval notation, this is $(-\infty, 2) \cup (2, +\infty)$.
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