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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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Question

Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?

Sum
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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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Chapter 16: Increasing and Decreasing Functions - Exercise 17.1 [Page 10]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 16 Increasing and Decreasing Functions
Exercise 17.1 | Q 2 | Page 10

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