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

Answer 1

To simplify the expression, we need to use logarithmic properties and rules.

First, let's simplify the expression inside the parentheses: log_7(8) × log_8(7)

We can use the change of base formula to express log_8(7) in terms of log_7(7):

log_8(7) = log_7(7) / log_7(8)

Now substitute this value back into the expression:

log_7(8) × log_8(7) = log_7(8) × (log_7(7) / log_7(8))

The log_7(8) terms cancel out, leaving us with:

log_7(8) × log_8(7) = log_7(7)

Now let's simplify the expression further:

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

Using the property of logarithms, log_b(b) = 1, we can simplify the first part of the expression:

7^(-log_7(7)) = 7^(log_7(7)^-1) = 7^(-1) = 1/7

So now we have:

(1/7) × 7 log_7(2)

Using the property of logarithms, log_b(b^x) = x, we can simplify the second part of the expression:

7 log_7(2) = log_7(2^7) = log_7(128)

Our simplified expression is now:

(1/7) × log_7(128)

And that's the final answer.