To simplify this expression, we start by using the properties of logarithms to simplify the exponents.
1. Let's start with the exponent in the base of 7.
log_7(8) can be rewritten as log(8)/log(7) using the change of base formula.
log_8(7) can be rewritten as log(7)/log(8) using the change of base formula.
2. Now, we substitute these values back into the expression.
7^(-(log(8)/log(7) * log(7)/log(8))) * 7 * log_7(2)
= 7^(-log(8)/log(8)) * 7 * log_7(2)
3. The exponent in the base of 7 can be rewritten using the property that a^(-b) = 1/(a^b).
= 1/(7^(log(8)/log(8))) * 7 * log_7(2)
4. Since 7^(log(8)/log(8)) = (7^(log(8)))^(1/log(8)) = 8^(1/log(8)), we can rewrite the expression as:
= 1/(8^(1/log(8))) * 7 * log_7(2)
5. Finally, we use the fact that log_a(b) = log_c(b)/log_c(a) to rewrite log_8(2) using a different base.
log_8(2) = log(2)/log(8) = log(2)/log(2^3) = log(2)/3log(2) = 1/3.
6. Therefore, the final expression becomes:
= 1/(8^(1/3)) * 7 * log_7(2)
= 1/2 * 7 * log_7(2)
= 7/2 * log_7(2)
7 ^ (- (log_7(8) * log_8(7))) * 7 * log_7(2)
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