7 ^ - (log_7(8) × log_8(7) × 7 log_7(2)

Answer 1

To simplify this expression, we can use the properties of logarithms and exponents.

First, let's rewrite the expression with parentheses to indicate the order of operations:
7^(- (log_7(8) × log_8(7) × 7 log_7(2)))

Now let's simplify each part step by step:
1. log_7(8) can be rewritten as log(8)/log(7) using the change of base formula.
2. log_8(7) can be rewritten as log(7)/log(8) using the change of base formula.
3. 7 log_7(2) can be rewritten as log(2) using the property log_b(b^x) = x.

So, the expression becomes:
7^(-(log(8)/log(7) × log(7)/log(8) × log(2)))

Since log(a) - log(b) = log(a/b), we can simplify further:
7^(-(log(2)))

Finally, we know that a^(-b) = 1/(a^b), so the expression becomes:
1 / 7^log(2)

Therefore, the simplified expression is 1 / 7^log(2).