Calculate `7^(log g_(3)5) + 3 ^(log_(5)7 ) - 5 ^(log_(3)7) - 7 ^(log_(5)3)`

Options

A. 1
B. 0
C. `7^(log_(3)5`
D. None

Answer

To solve the expression \( 7^{\log_3 5} + 3^{\log_5 7} - 5^{\log_3 7} - 7^{\log_5 3} \), we will use the logarithmic identity:

\[ a^{\log_b c} = c^{\log_b a} \]

### Step 1: Rewrite the terms using the logarithmic identity

1. **First Term**:
\[ 7^{\log_3 5} = 5^{\log_3 7} \]
(Using the identity with \( a = 7, b = 3, c = 5 \))

2. **Second Term**:
\[ 3^{\log_5 7} = 7^{\log_5 3} \]
(Using the identity with \( a = 3, b = 5, c = 7 \))

### Step 2: Substitute back into the expression

Now substituting these back into the original expression:

\[ 5^{\log_3 7} + 7^{\log_5 3} - 5^{\log_3 7} - 7^{\log_5 3} \]

### Step 3: Simplify the expression

Notice that \( 5^{\log_3 7} \) and \( -5^{\log_3 7} \) cancel each other out, and similarly \( 7^{\log_5 3} \) and \( -7^{\log_5 3} \) also cancel each other out:

\[ 0 \]

### Final Result

Thus, the final result is:

\[ \boxed{0} \]

---