7 ^ - (log_7(8) × log_8(7) * 7 log_7(2)
To simplify the expression, let's break it down step by step:
Step 1: Simplify the logarithmic terms.
Since log base b (b) = 1, we have log_7(7) = 1 and log_8(8) = 1.
Step 2: Substitute the values into the expression.
7^- (log_7(8) × log_8(7) * 7 log_7(2))
= 7^- (1 × 1 * 7 log_7(2))
Step 3: Simplify the exponent expression.
7 log_7(2) = 2 (as 7^2 = 49)
So, the expression becomes:
7^-(2) = 1/7^2 = 1/49
Therefore, the simplified expression is 1/49.
To solve this expression, we need to simplify each term individually and then perform the operations.
Let's simplify each term step by step:
1) log_7(8) can be written as log(8) / log(7) using the change of base formula. We can use a calculator to calculate the value of log(8) and log(7) separately.
log(8) ≈ 0.9031
log(7) ≈ 0.8451
So, log_7(8) ≈ 0.9031 / 0.8451 ≈ 1.0687
2) log_8(7) can be written as log(7) / log(8).
log(7) ≈ 0.8451
log(8) ≈ 0.9031
So, log_8(7) ≈ 0.8451 / 0.9031 ≈ 0.9360
3) 7 log_7(2) can be simplified using the property: a log_a(p) = p.
So, 7 log_7(2) = 7*(2) = 14.
Now, let's substitute these simplified values back into the original expression:
7^(- (log_7(8) × log_8(7) * 7 log_7(2))
= 7^(- (1.0687 × 0.9360 * 14))
= 7^(- (14.0166))
≈ 0.000000004404
Therefore, 7^(- (log_7(8) × log_8(7) * 7 log_7(2)) ≈ 0.000000004404.
To find the value of the expression, let's break it down step by step:
1. Start with log_7(8) × log_8(7).
- Use the property of logarithms: log_a(b) = 1 / log_b(a).
- So, log_7(8) × log_8(7) = (1 / log_8(7)) × (1 / log_7(8)).
- Multiplying these fractions gives us 1 / (log_8(7) × log_7(8)).
2. Now we have 7 ^ - ( 1 / (log_8(7) × log_7(8)) ) * 7 log_7(2).
- Simplify 7 ^ - ( 1 / (log_8(7) × log_7(8)) ).
- This is equivalent to 7 ^ -1 * (log_8(7) × log_7(8)).
- Since 7 ^ -1 is the reciprocal of 7, it becomes 1/7.
3. The expression is now (1/7) * 7 log_7(2).
- When we multiply 1/7 by 7, they cancel out, leaving us with log_7(2).
Therefore, the value of the expression 7 ^ - (log_7(8) × log_8(7) * 7 log_7(2)) is equal to log_7(2).