The value of `(2^(log3^(7) - 7^(log _(3)2)))` is

Options

A. 0
B. 1
C. `log_(7)5`
D. `log_(7)6`

Answer

To solve the expression \( 2^{(\log_3 7 - 7^{(\log_3 2)})} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 2^{(\log_3 7 - 7^{(\log_3 2)})} \]

### Step 2: Apply the logarithmic identity
Using the property of logarithms, we know that:
\[ a^{\log_b c} = c^{\log_b a} \]
We can rewrite \( 7^{(\log_3 2)} \) as:
\[ 7^{(\log_3 2)} = 2^{(\log_3 7)} \]
Thus, we can rewrite our expression as:
\[ 2^{(\log_3 7 - 2^{(\log_3 7)})} \]

### Step 3: Simplify the expression
Now, we have:
\[ \log_3 7 - 2^{(\log_3 7)} \]
This means we have:
\[ \log_3 7 - \log_3 7 = 0 \]
So, the expression simplifies to:
\[ 2^0 \]

### Step 4: Final calculation
Since \( 2^0 = 1 \), we conclude that:
\[ 2^{(\log_3 7 - 7^{(\log_3 2)})} = 1 \]

### Conclusion
The value of \( 2^{(\log_3 7 - 7^{(\log_3 2)})} \) is \( 1 \).

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