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# Solution for Write the Set of Values of 'A' for Which F(X) = Loga X is Decreasing in Its Domain ? - CBSE (Commerce) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

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

#### Solution

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

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

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

$\text { Since the given function is logarithmic, eithera } > 1 or0 < a < 1 .$

$\text { Case 1: Let }a > 1$

$\text { Here} ,$

$x_1 < x_2$

$\Rightarrow \log_a x_1 < \log_a x_2$

$\Rightarrow f\left( x_1 \right) < f\left( x_2 \right)$

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

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

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

$\text { Here, }$

$x_1 < x_2$

$\Rightarrow \log_a x_1 > \log_a x_2$

$\Rightarrow f\left( x_1 \right) > f\left( x_2 \right)$

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

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

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

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Solution Write the Set of Values of 'A' for Which F(X) = Loga X is Decreasing in Its Domain ? Concept: Increasing and Decreasing Functions.
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