If `0 lt a lt 1`, then the solution set of the inequation
`(1+(log_(a)x)^(2))/(1+(log_(a)x)) gt1`, is

Options

A. `(1,1//a)`
B. (0,a)
C. `(1,1//a) cup (0,a)`
D. None of these

Answer

To solve the inequality

\[ \frac{1 + (\log_a x)^2}{1 + \log_a x} > 1 \]

where \(0 < a < 1\), we will follow these steps:

### Step 1: Rewrite the Inequality
Start by rewriting the inequality:

\[ \frac{1 + (\log_a x)^2}{1 + \log_a x} - 1 > 0 \]

This simplifies to:

\[ \frac{1 + (\log_a x)^2 - (1 + \log_a x)}{1 + \log_a x} > 0 \]

### Step 2: Simplify the Numerator
Now simplify the numerator:

\[ 1 + (\log_a x)^2 - 1 - \log_a x = (\log_a x)^2 - \log_a x \]

Thus, the inequality becomes:

\[ \frac{(\log_a x)^2 - \log_a x}{1 + \log_a x} > 0 \]

### Step 3: Factor the Numerator
Factor the numerator:

\[ \frac{\log_a x (\log_a x - 1)}{1 + \log_a x} > 0 \]

### Step 4: Determine Critical Points
Identify the critical points by setting the numerator and denominator to zero:

1. \(\log_a x = 0 \Rightarrow x = a^0 = 1\)
2. \(\log_a x - 1 = 0 \Rightarrow x = a^1 = a\)
3. \(1 + \log_a x = 0 \Rightarrow \log_a x = -1 \Rightarrow x = a^{-1} = \frac{1}{a}\)

### Step 5: Analyze the Sign of the Expression
We need to analyze the sign of the expression \(\frac{\log_a x (\log_a x - 1)}{1 + \log_a x}\) across the intervals determined by the critical points \(x = a\), \(x = 1\), and \(x = \frac{1}{a}\).

- **Interval 1:** \(x < a\)
- \(\log_a x < 0\) (negative)
- \(\log_a x - 1 < 0\) (negative)
- \(1 + \log_a x > 0\) (positive)

The product is positive.

- **Interval 2:** \(a < x < 1\)
- \(\log_a x > 0\) (positive)
- \(\log_a x - 1 < 0\) (negative)
- \(1 + \log_a x > 0\) (positive)

The product is negative.

- **Interval 3:** \(x = 1\)
- The expression equals zero.

- **Interval 4:** \(1 < x < \frac{1}{a}\)
- \(\log_a x > 0\) (positive)
- \(\log_a x - 1 > 0\) (positive)
- \(1 + \log_a x > 0\) (positive)

The product is positive.

- **Interval 5:** \(x > \frac{1}{a}\)
- \(\log_a x > 0\) (positive)
- \(\log_a x - 1 > 0\) (positive)
- \(1 + \log_a x > 0\) (positive)

The product is positive.

### Step 6: Combine the Results
From the analysis, the solution set is:

\[ (-\infty, a) \cup (1, \frac{1}{a}) \]

### Final Answer
Thus, the solution set of the inequality is:

\[ x \in (-\infty, a) \cup (1, \frac{1}{a}) \]