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Question
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
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Solution
\[\text { Given }: f\left( x \right) = \log_a x\]
\[\text { Domain of the given function is }\left( 0, \infty \right).\]
\[\text { Let }x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \]
\[\text { Since the given function is logarithmic, either a } > 1 or 0 < a < 1 . \]
\[\text { Case 1: Let }a > 1\]
\[\text { Here} , \]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 < \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So },f\left( x \right)\text { is increasing on }\left( 0, \infty \right).\]
\[\text { Case 2: Let }0 < a < 1\]
\[\text { Here, }\]
\[ x_1 < x_2 \]
\[ \Rightarrow \log_a x_1 > \log_a x_2 \]
\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]
\[\text { So,}f\left( x \right) \text { is decreasing on }\left( 0, \infty \right)\]
\[\text { Thus, for }0 < a < 1,f\left( x \right)\text { is decreasing in its domain }.\]
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