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Question
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
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Solution
\[f\left( x \right) = \log_a x\]
\[\text { Let } x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \]
\[\text { Case 1: Let a } > 1\]
\[\text{ Here },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 < \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[ \therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So,}f\left( x \right) \text { is increasing on }\left( 0, \infty \right).\]
\[\text { Case 2: Let }0 < a < 1\]
\[\text { Here },\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 > \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[ \therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So },f\left( x \right)\text { is decreasing on }\left( 0, \infty \right).\]
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