Statement-1(Assertion) and Statement-2 (reason) Each of these question also has four alternative choices, only one of which is the correct answer. You have to select the correct choice as given below.
(a) Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1
Statement-1 is true, Statement-2 is true, Statement-2 is not a correct explanation for Statement -1
(c) Statement -1 is true, Statement -2 is false
(d) Statement -1 is false, Statement -2 is true
Statement -1 `log_x3.log_(x//9)3=log_81(3)` has a solution.
Statement-2 Change of base in logarithms is possible .

Answer

To solve the problem, we need to analyze the two statements provided:

**Statement 1:** \( \log_x 3 \cdot \log_{\frac{x}{9}} 3 = \log_{81} 3 \) has a solution.

**Statement 2:** Change of base in logarithms is possible.

### Step-by-Step Solution:

1. **Understanding Statement 2:**
- Statement 2 is true because the change of base formula for logarithms allows us to convert logarithms from one base to another. The formula is:
\[ \log_a b = \frac{\log_c b}{\log_c a} \]
- This means we can express logarithms in any base we choose, confirming that Statement 2 is true.

**Hint:** Remember the change of base formula for logarithms.

2. **Analyzing Statement 1:**
- We need to check if \( \log_x 3 \cdot \log_{\frac{x}{9}} 3 = \log_{81} 3 \) has a solution.
- Using the change of base formula, we can rewrite the logarithms:
\[ \log_x 3 = \frac{\log 3}{\log x} \]
\[ \log_{\frac{x}{9}} 3 = \frac{\log 3}{\log \frac{x}{9}} = \frac{\log 3}{\log x - \log 9} \]
\[ \log_{81} 3 = \frac{\log 3}{\log 81} \]
- Substituting these into Statement 1 gives:
\[ \frac{\log 3}{\log x} \cdot \frac{\log 3}{\log x - \log 9} = \frac{\log 3}{\log 81} \]

3. **Simplifying the Equation:**
- Let's simplify the left side:
\[ \frac{(\log 3)^2}{\log x (\log x - \log 9)} = \frac{\log 3}{\log 81} \]
- Cross-multiplying gives:
\[ (\log 3)^2 \cdot \log 81 = \log 3 \cdot \log x (\log x - \log 9) \]
- Dividing both sides by \( \log 3 \) (assuming \( \log 3 \neq 0 \)):
\[ \log 3 \cdot \log 81 = \log x (\log x - \log 9) \]

4. **Finding the Values:**
- We know that \( \log 81 = 4 \log 3 \) (since \( 81 = 3^4 \)):
\[ \log 3 \cdot 4 \log 3 = \log x (\log x - \log 9) \]
- This simplifies to:
\[ 4 (\log 3)^2 = \log x (\log x - 2 \log 3) \]
- Let \( \log x = t \):
\[ 4 t^2 = t(t - 2 \log 3) \]
- Rearranging gives:
\[ 4t^2 - t^2 + 2t \log 3 = 0 \]
\[ 3t^2 + 2t \log 3 = 0 \]

5. **Solving the Quadratic Equation:**
- Factoring out \( t \):
\[ t(3t + 2 \log 3) = 0 \]
- This gives \( t = 0 \) or \( 3t + 2 \log 3 = 0 \).
- The first solution \( t = 0 \) implies \( x = 10^0 = 1 \), which is valid.
- The second solution gives \( t = -\frac{2 \log 3}{3} \), which leads to a negative logarithm, hence not valid.

6. **Conclusion:**
- Since Statement 1 does not have a valid solution (as \( x = 1 \) does not satisfy the original logarithmic equation), we conclude that:
- Statement 1 is false.
- Statement 2 is true.

### Final Answer:
- The correct choice is (d): Statement 1 is false, Statement 2 is true.