Question

Write the set of values of 'a' for which f(x) = log_a x is decreasing in its domain ?

Answer 1

\[\text { Given }: f\left( x \right) = \log_a x\]

\[\text { Domain of the given function is }\left( 0, \infty \right).\]

\[\text { Let  }x_1 , x_2 \in \left( 0, \infty \right) \text { such that } x_1 < x_2 . \]

\[\text { Since the given function is logarithmic, either a } > 1 or 0 < a < 1 . \]

\[\text { Case 1: Let }a > 1\]

\[\text { Here} , \]

\[ x_1 < x_2 \]

\[ \Rightarrow \log_a x_1 < \log_a x_2 \]

\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]

\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]

\[\text { So },f\left( x \right)\text { is increasing on }\left( 0, \infty \right).\]

\[\text { Case 2: Let }0 < a < 1\]

\[\text { Here, }\]

\[ x_1 < x_2 \]

\[ \Rightarrow \log_a x_1 > \log_a x_2 \]

\[ \Rightarrow f\left( x_1 \right) > f\left( x_2 \right)\]

\[\therefore x_1 < x_2 \Rightarrow f\left( x_1 \right) > f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \infty \right)\]

\[\text { So,}f\left( x \right) \text { is decreasing on }\left( 0, \infty \right)\]

\[\text { Thus, for }0 < a < 1,f\left( x \right)\text {  is decreasing in its domain }.\]