The number of integral solutions of `log_(9)(x+1).log_(2)(x+1)-log_(9)(x+1)-log_(2)(x+1)+1lt0` is

Options

A. 4
B. 5
C. 6
D. 7

Explanation

To solve the inequality \( \log_{9}(x+1) \cdot \log_{2}(x+1) - \log_{9}(x+1) - \log_{2}(x+1) + 1 < 0 \), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the given inequality:
\[ \log_{9}(x+1) \cdot \log_{2}(x+1) - \log_{9}(x+1) - \log_{2}(x+1) + 1 < 0 \]
We can rearrange it as:
\[ \log_{9}(x+1) \cdot \log_{2}(x+1) - \log_{9}(x+1) - \log_{2}(x+1) < -1 \]

### Step 2: Factor the Left Side
Next, we factor the left-hand side:
\[ (\log_{9}(x+1) - 1)(\log_{2}(x+1) - 1) < 0 \]
This means we need to analyze the product of two terms.

### Step 3: Set Each Term to Zero
Now, we find the points where each term is zero:
1. \( \log_{9}(x+1) - 1 = 0 \) implies \( \log_{9}(x+1) = 1 \) which gives \( x + 1 = 9 \) or \( x = 8 \).
2. \( \log_{2}(x+1) - 1 = 0 \) implies \( \log_{2}(x+1) = 1 \) which gives \( x + 1 = 2 \) or \( x = 1 \).

### Step 4: Analyze Intervals
Now we analyze the intervals determined by the roots \( x = 1 \) and \( x = 8 \):
- Interval 1: \( (-\infty, 1) \)
- Interval 2: \( (1, 8) \)
- Interval 3: \( (8, \infty) \)

### Step 5: Test Each Interval
We will test a point in each interval to see where the product is negative:

1. **For \( x < 1 \)** (e.g., \( x = 0 \)):
- \( \log_{9}(1) - 1 = -1 \) (negative)
- \( \log_{2}(1) - 1 = -1 \) (negative)
- Product: Positive

2. **For \( 1 < x < 8 \)** (e.g., \( x = 2 \)):
- \( \log_{9}(3) - 1 < 0 \) (negative)
- \( \log_{2}(3) - 1 > 0 \) (positive)
- Product: Negative

3. **For \( x > 8 \)** (e.g., \( x = 9 \)):
- \( \log_{9}(10) - 1 > 0 \) (positive)
- \( \log_{2}(10) - 1 > 0 \) (positive)
- Product: Positive

### Step 6: Determine the Solution Set
The inequality \( (\log_{9}(x+1) - 1)(\log_{2}(x+1) - 1) < 0 \) holds true in the interval \( (1, 8) \).

### Step 7: Find Integral Solutions
The integral solutions in the interval \( (1, 8) \) are:
\[ x = 2, 3, 4, 5, 6, 7 \]
Thus, there are a total of 6 integral solutions.

### Final Answer
The number of integral solutions is \( \boxed{6} \).