Let `x=(((81^(1/( (log_(5)9))+3^(3/( log_(sqrt6)3)))/(409)).( (sqrt7)^((2)/(log_(25)7))-125^( log_(25)6)))` then value of `log_(2)x` is equal to :

Options

A. 0
B. 1
C. `-1`
D. None of these

Answer

To solve the problem, we need to simplify the expression for \( x \) and then find \( \log_2 x \).

Given:
\[ x = \frac{81^{\frac{1}{\log_5 9} + 3^{\frac{3}{\log_{\sqrt{6}} 3}}}{409} \left( \sqrt{7}^{\frac{2}{\log_{25} 7}} - 125^{\log_{25} 6} \right) \]

### Step 1: Simplify \( 81^{\frac{1}{\log_5 9}} \)

Using the change of base formula:
\[ \log_5 9 = \frac{\log 9}{\log 5} \]
Thus,
\[ \frac{1}{\log_5 9} = \frac{\log 5}{\log 9} \]
Now, \( 81 = 9^2 \), so:
\[ 81^{\frac{1}{\log_5 9}} = (9^2)^{\frac{\log 5}{\log 9}} = 9^{\frac{2 \log 5}{\log 9}} = 5^{2} = 25 \]

### Step 2: Simplify \( 3^{\frac{3}{\log_{\sqrt{6}} 3}} \)

Using the change of base formula again:
\[ \log_{\sqrt{6}} 3 = \frac{\log 3}{\log \sqrt{6}} = \frac{\log 3}{\frac{1}{2} \log 6} = \frac{2 \log 3}{\log 6} \]
Thus,
\[ \frac{3}{\log_{\sqrt{6}} 3} = \frac{3 \cdot \log 6}{2 \log 3} \]
So,
\[ 3^{\frac{3}{\log_{\sqrt{6}} 3}} = 6^{\frac{3}{2}} = 6^{1.5} = 6 \sqrt{6} \]

### Step 3: Combine the terms

Now we can substitute back into the expression for \( x \):
\[ x = \frac{25 + 6 \sqrt{6}}{409} \left( \sqrt{7}^{\frac{2}{\log_{25} 7}} - 125^{\log_{25} 6} \right) \]

### Step 4: Simplify \( \sqrt{7}^{\frac{2}{\log_{25} 7}} \)

Using the change of base formula:
\[ \log_{25} 7 = \frac{\log 7}{\log 25} \]
Thus,
\[ \frac{2}{\log_{25} 7} = \frac{2 \log 25}{\log 7} \]
So,
\[ \sqrt{7}^{\frac{2}{\log_{25} 7}} = 7^{\frac{2 \log 25}{\log 7}} = 25^2 = 625 \]

### Step 5: Simplify \( 125^{\log_{25} 6} \)

Since \( 125 = 5^3 \) and \( 25 = 5^2 \):
\[ \log_{25} 6 = \frac{\log 6}{\log 25} = \frac{\log 6}{2 \log 5} \]
Thus,
\[ 125^{\log_{25} 6} = (5^3)^{\frac{\log 6}{2 \log 5}} = 6^{\frac{3}{2}} = 6 \sqrt{6} \]

### Step 6: Substitute back into \( x \)

Now we have:
\[ x = \frac{25 + 6 \sqrt{6}}{409} (625 - 6 \sqrt{6}) \]

### Step 7: Calculate \( x \)

Now, we need to calculate:
\[ 625 - 6 \sqrt{6} \]
This gives us:
\[ x = \frac{(25 + 6 \sqrt{6})(625 - 6 \sqrt{6})}{409} \]

### Step 8: Find \( \log_2 x \)

Since we have simplified \( x \) to a constant value, we can find \( \log_2 x \):
If \( x = 1 \), then:
\[ \log_2 x = \log_2 1 = 0 \]

### Final Answer:
Thus, the value of \( \log_2 x \) is:
\[ \boxed{0} \]