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 given expression for \( x \) and find the value of \( \log_2 x \), we will break it down step by step.

### Step 1: Rewrite the expression for \( x \)

The expression for \( x \) is given as:

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

### Step 2: Simplify \( 81^{\frac{1}{\log_5 9}} \)

Using the property \( a^{\frac{1}{\log_b a}} = b \):

\[ 81^{\frac{1}{\log_5 9}} = 9^{\log_5 81} \]

Since \( 81 = 9^2 \):

\[ 9^{\log_5 81} = 9^{\log_5 (9^2)} = 9^{2 \log_5 9} = 5^2 = 25 \]

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

Using the same property:

\[ 3^{\frac{3}{\log_{\sqrt{6}} 3}} = 3^{3 \cdot \log_3 \sqrt{6}} = 6^{\frac{3}{2}} = 6\sqrt{6} \]

### Step 4: Combine the results

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 5: Simplify \( \sqrt{7}^{\frac{2}{\log_{25} 7}} \)

Using the property again:

\[ \sqrt{7}^{\frac{2}{\log_{25} 7}} = 7^{\log_{25} \sqrt{7}} = 25 \]

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

Since \( 125 = 5^3 \):

\[ 125^{\log_{25} 6} = 5^{3 \log_{25} 6} = 6^{\frac{3}{2}} = 6\sqrt{6} \]

### Step 7: Substitute back into \( x \)

Now we can substitute these results back into the expression for \( x \):

\[ x = \frac{25 + 6\sqrt{6}}{409} \left( 25 - 6\sqrt{6} \right) \]

### Step 8: Calculate the final value of \( x \)

Calculating the expression inside the parentheses:

\[ x = \frac{(25 + 6\sqrt{6})(25 - 6\sqrt{6})}{409} \]

Using the difference of squares:

\[ = \frac{25^2 - (6\sqrt{6})^2}{409} = \frac{625 - 216}{409} = \frac{409}{409} = 1 \]

### Step 9: Find \( \log_2 x \)

Finally, we find:

\[ \log_2 x = \log_2 1 = 0 \]

### Final Answer:

The value of \( \log_2 x \) is \( 0 \).

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