Question
Show that $\text{f}(\text{x})=\log_{\text{a}}\text{x},0<\text{a}<1$ is a decreasing function for all x > 0.
✓
Answer
$\text{f}(\text{x})=\log_{\text{a}}\text{x}$
$=\frac{\log\text{x}}{\log\text{a}}$
$\text{f}'(\text{x})=\frac{1}{\text{x}\log\text{a}}$
Since, $0<\text{a}<1$ and $\text{f}'(\text{x})=\frac{1}{\text{x}\log\text{a}}<0.$
Hence, f(x) is decreasing function for all x > 0.
$=\frac{\log\text{x}}{\log\text{a}}$
$\text{f}'(\text{x})=\frac{1}{\text{x}\log\text{a}}$
Since, $0<\text{a}<1$ and $\text{f}'(\text{x})=\frac{1}{\text{x}\log\text{a}}<0.$
Hence, f(x) is decreasing function for all x > 0.
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