Prove that the function `f(x)=(log)_a x` is increasing on `(0,\ oo)` if `a >1` and decreasing on `(0,\ oo)` . if `0

Explanation

To prove that the function \( f(x) = \log_a x \) is increasing on \( (0, \infty) \) if \( a > 1 \) and decreasing on \( (0, \infty) \) if \( 0 < a < 1 \), we will follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function \( f(x) \). The derivative of \( f(x) = \log_a x \) can be expressed using the change of base formula:

\[ f'(x) = \frac{d}{dx} \left( \frac{\log x}{\log a} \right) = \frac{1}{\log a} \cdot \frac{1}{x} \]
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