What are the values of ''a for which the function `f(x)=log_(a)x` is :
(i) increasing
(ii) decreasing in its domain?

Answer

To determine the values of \( a \) for which the function \( f(x) = \log_a x \) is increasing or decreasing in its domain, we will analyze the behavior of the logarithmic function based on the value of \( a \).

### Step 1: Understand the Domain of \( f(x) \)
The function \( f(x) = \log_a x \) is defined for \( x > 0 \) and \( a > 0 \), \( a \neq 1 \).

### Step 2: Determine the Derivative
To find when the function is increasing or decreasing, we need to find the derivative of \( f(x) \):
\[ f'(x) = \frac{1}{x \ln a} \]
Here, \( \ln a \) is the natural logarithm of \( a \).

### Step 3: Analyze the Sign of the Derivative
The sign of \( f'(x) \) determines whether the function is increasing or decreasing:
- **Increasing**: \( f'(x) > 0 \)
- **Decreasing**: \( f'(x) < 0 \)

### Step 4: Case Analysis Based on \( a \)
1. **Case 1: \( a > 1 \)**
- Here, \( \ln a > 0 \).
- Therefore, \( f'(x) = \frac{1}{x \ln a} > 0 \) for all \( x > 0 \).
- Conclusion: \( f(x) \) is increasing when \( a > 1 \).

2. **Case 2: \( 0 < a < 1 \)**
- Here, \( \ln a < 0 \).
- Therefore, \( f'(x) = \frac{1}{x \ln a} < 0 \) for all \( x > 0 \).
- Conclusion: \( f(x) \) is decreasing when \( 0 < a < 1 \).

3. **Case 3: \( a = 1 \)**
- The function \( f(x) = \log_1 x \) is undefined, so we do not consider this case.

### Final Conclusions
- The function \( f(x) = \log_a x \) is:
- **Increasing** for \( a > 1 \)
- **Decreasing** for \( 0 < a < 1 \)