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Question
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
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Solution
\[f\left( x \right) = \log_a x\]
\[ = \frac{\log x}{\log a}\]
\[f'\left( x \right) = \frac{1}{x \log a}\]
\[\text { Since 0 < a < 1 and } x > 0, f'\left( x \right) = \frac{1}{x \log a} < 0 . \]
\[\text { So,}f\left( x \right) \text { is decreasing for all } x > 0 .\]
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