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

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Question

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

Solution

\[f\left( x \right) = \log_a x\]

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

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

\[\text { Case 1: Let }a  > 1\]

\[\text { Here },\]

\[ x_1 < x_2 \]

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

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

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

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

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

\[\text { Here },\]

\[ x_1 < x_2 \]

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

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

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

\[\text { Thus, for } a > 1, f(x)\text {  is increasing in its domain } . \]

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Solution for question: Write the Set of Values of 'A' for Which F(X) = Loga X is Increasing in Its Domain ? concept: Increasing and Decreasing Functions. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
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