The value of `(1+2(log_3 2)/((1+(log)_3 2)^2)+((log)_6 2)^2` is 2 (b) 3 (c) 4 (d) 1

Options

A. 2
B. 3
C. 4
D. 1

Explanation

To solve the expression \( 1 + \frac{2 \log_3 2}{(1 + \log_3 2)^2} + (\log_6 2)^2 \), we will break it down step by step.

### Step 1: Rewrite \(\log_6 2\)
We can use the change of base formula to rewrite \(\log_6 2\):
\[ \log_6 2 = \frac{\log_3 2}{\log_3 6} \]
Since \(\log_3 6 = \log_3 (3 \cdot 2) = \log_3 3 + \log_3 2 = 1 + \log_3 2\), we have:
\[ \log_6 2 = \frac{\log_3 2}{1 + \log_3 2} \]

### Step 2: Substitute \(\log_6 2\) into the expression
Now substitute \(\log_6 2\) back into the original expression:
\[ 1 + \frac{2 \log_3 2}{(1 + \log_3 2)^2} + \left(\frac{\log_3 2}{1 + \log_3 2}\right)^2 \]

### Step 3: Simplify the expression
Let \(x = \log_3 2\). The expression now becomes:
\[ 1 + \frac{2x}{(1 + x)^2} + \left(\frac{x}{1 + x}\right)^2 \]

### Step 4: Simplify \(\left(\frac{x}{1 + x}\right)^2\)
Calculating \(\left(\frac{x}{1 + x}\right)^2\):
\[ \left(\frac{x}{1 + x}\right)^2 = \frac{x^2}{(1 + x)^2} \]

### Step 5: Combine the terms
Now, we can combine the terms:
\[ 1 + \frac{2x}{(1 + x)^2} + \frac{x^2}{(1 + x)^2} = 1 + \frac{2x + x^2}{(1 + x)^2} \]
This simplifies to:
\[ 1 + \frac{x^2 + 2x}{(1 + x)^2} = 1 + \frac{(x + 1)^2}{(1 + x)^2} \]

### Step 6: Final simplification
The term \(\frac{(x + 1)^2}{(1 + x)^2}\) simplifies to \(1\):
\[ 1 + 1 = 2 \]

### Conclusion
Thus, the value of the expression is:
\[ \boxed{2} \]