7 ^ - (log_7(8) × log_8(7) × 7 log_7(2)
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).
To simplify the given expression, we need to understand the properties of exponents and logarithms.
Let's break down the expression step by step:
1. Start by simplifying the logarithmic terms: log_7(8) and log_8(7). We can use the formula: log_a(b) = log_c(b) / log_c(a), where c is any base.
log_7(8) = log_10(8) / log_10(7)
log_8(7) = log_10(7) / log_10(8)
2. Multiply these two logarithmic terms:
log_7(8) × log_8(7) = (log_10(8) / log_10(7)) × (log_10(7) / log_10(8))
= (log_10(8) × log_10(7)) / (log_10(7) × log_10(8))
= 1
Therefore, log_7(8) × log_8(7) simplifies to 1.
3. The expression now becomes:
7^-(1) × (7 log_7(2))
Since 7^-(1) is equal to 1/7, we have:
(1/7) × (7 log_7(2))
4. The term (7 log_7(2)) can be simplified using the logarithmic property: log_x(x) = 1. In this case, log_7(7) is 1.
(1/7) × 7 × log_7(2)
= 1 × log_7(2)
5. Finally, the expression becomes:
log_7(2)
Therefore, the simplified expression is log_7(2).
To simplify the expression 7 ^ - (log_7(8) × log_8(7) × 7 log_7(2), let's break it down step-by-step:
Step 1: Solve the logarithmic terms:
- log_7(8) = x
This equation can be rewritten as 7^x = 8.
Since we know that 8 is 2^3, we can rewrite the equation as 7^x = 2^3.
Now, both sides have the same base, so the exponents must be equal: x = 3.
- log_8(7) = y
This equation can be rewritten as 8^y = 7.
There is no simple way to solve this equation analytically, but we can estimate the value. By trial and error, we can find that y is approximately 0.876.
- log_7(2) = z
This equation can be rewritten as 7^z = 2.
Again, there is no simple way to solve this equation analytically, but we can estimate the value. By trial and error, we can find that z is approximately 0.509.
Step 2: Substitute the values of x, y, and z into the expression:
7 ^ - (log_7(8) × log_8(7) × 7 log_7(2)
= 7 ^ - (3 × 0.876 × 7 × 0.509)
= 7^-11.753
Step 3: Evaluate the expression:
7^-11.753 is equivalent to taking the reciprocal of 7^11.753.
So, 7^-11.753 = 1 / 7^11.753.
Therefore, the simplified form of the expression is 1 / 7^11.753.