If `log_(4)A=log_(6)B=log_(9)(A+B)," then "[4(B//A)]("where "[*]` represents the greatest integer function ) equals _______.

Explanation

To solve the problem given, we need to find the value of \([4(B/A)]\) where \([*]\) represents the greatest integer function, given that \( \log_{4} A = \log_{6} B = \log_{9} (A + B) \).

### Step-by-Step Solution:

1. **Set the common logarithm value**:
Let \( x = \log_{4} A = \log_{6} B = \log_{9} (A + B) \).

2. **Express A and B in terms of x**:
From the properties of logarithms:
\[ A = 4^x \]
\[ B = 6^x \]

3. **Express A + B**:
Using the values of A and B:
\[ A + B = 4^x + 6^x \]

4. **Express A + B in terms of logarithm**:
Since \( \log_{9} (A + B) = x \), we can express \( A + B \) as:
\[ A + B = 9^x \]

5. **Set up the equation**:
Now we have:
\[ 4^x + 6^x = 9^x \]

6. **Rewrite the equation**:
We can rewrite \( 4^x \) and \( 9^x \) in terms of base 2 and base 3:
\[ (2^2)^x + (2 \cdot 3)^x = (3^2)^x \]
This simplifies to:
\[ 2^{2x} + 2^x \cdot 3^x = 3^{2x} \]

7. **Divide through by \( 2^{2x} \)**:
\[ 1 + \frac{3^x}{2^x} = \left(\frac{3}{2}\right)^{2x} \]

8. **Let \( t = \left(\frac{3}{2}\right)^x \)**:
Then the equation becomes:
\[ 1 + t = t^2 \]
Rearranging gives:
\[ t^2 - t - 1 = 0 \]

9. **Solve the quadratic equation**:
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{1 \pm \sqrt{5}}{2} \]
We take the positive root:
\[ t = \frac{1 + \sqrt{5}}{2} \]

10. **Substitute back to find A and B**:
Since \( t = \left(\frac{3}{2}\right)^x \):
\[ \left(\frac{3}{2}\right)^x = \frac{1 + \sqrt{5}}{2} \]

11. **Find \( \frac{B}{A} \)**:
\[ \frac{B}{A} = \frac{6^x}{4^x} = \left(\frac{6}{4}\right)^x = \left(\frac{3}{2}\right)^x = t \]

12. **Calculate \( 4 \cdot \frac{B}{A} \)**:
\[ 4 \cdot \frac{B}{A} = 4t = 4 \cdot \frac{1 + \sqrt{5}}{2} = 2(1 + \sqrt{5}) = 2 + 2\sqrt{5} \]

13. **Approximate \( 2 + 2\sqrt{5} \)**:
Since \( \sqrt{5} \approx 2.236 \):
\[ 2 + 2\sqrt{5} \approx 2 + 4.472 = 6.472 \]

14. **Apply the greatest integer function**:
\[ [4(B/A)] = [6.472] = 6 \]

### Final Answer:
The value of \([4(B/A)]\) is \( \boxed{6} \).