Verify : `3^(sqrt((log_(3)7)))=7^(sqrt((log_(7)3)))`

Answer

To verify the equation \( 3^{\sqrt{\log_3 7}} = 7^{\sqrt{\log_7 3}} \), we will follow these steps:

### Step 1: Define Variables
Let:
- \( A = \sqrt{\log_3 7} \)
- \( B = \sqrt{\log_7 3} \)

### Step 2: Rewrite the Equation
We want to verify that:
\[ 3^A = 7^B \]
This can be rewritten as:
\[ \frac{A}{B} = \frac{\sqrt{\log_3 7}}{\sqrt{\log_7 3}} \]

### Step 3: Use Logarithmic Properties
Using the property of logarithms:
\[ \log_a b = \frac{1}{\log_b a} \]
We can express \( \log_7 3 \) in terms of \( \log_3 7 \):
\[ \log_7 3 = \frac{1}{\log_3 7} \]
Thus, we can rewrite \( B \):
\[ B = \sqrt{\log_7 3} = \sqrt{\frac{1}{\log_3 7}} = \frac{1}{\sqrt{\log_3 7}} \]

### Step 4: Substitute Back into the Ratio
Now substituting \( B \) back into our ratio:
\[ \frac{A}{B} = \frac{\sqrt{\log_3 7}}{\frac{1}{\sqrt{\log_3 7}}} = \sqrt{\log_3 7} \cdot \sqrt{\log_3 7} = \log_3 7 \]

### Step 5: Simplify the Expression
This simplifies to:
\[ \frac{A}{B} = \log_3 7 \]

### Step 6: Exponentiate Both Sides
Now, we exponentiate both sides to eliminate the logarithm:
\[ \log_3 7 = \frac{A}{B} \implies 7 = 3^{\frac{A}{B}} \]

### Step 7: Substitute Back to Original Variables
Since \( A = \sqrt{\log_3 7} \) and \( B = \sqrt{\log_7 3} \), we can write:
\[ 7^{\sqrt{\log_7 3}} = 3^{\sqrt{\log_3 7}} \]

### Conclusion
Thus, we have verified that:
\[ 3^{\sqrt{\log_3 7}} = 7^{\sqrt{\log_7 3}} \]
This confirms the original statement is true.