7 ^ - (log_7(8) × log_8(7)) × 7 log_7(2)
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.
To simplify the expression, let's break it down step by step:
Step 1: Solve log_7(8)
To solve log_7(8), we need to find the power to which 7 must be raised to get 8.
7^x = 8
Taking the logarithm of both sides with base 7:
x = log_7(8)
Step 2: Solve log_8(7)
To solve log_8(7), we need to find the power to which 8 must be raised to get 7.
8^y = 7
Taking the logarithm of both sides with base 8:
y = log_8(7)
Step 3: Calculate log_7(2)
To calculate log_7(2), we need to find the power to which 7 must be raised to get 2.
7^z = 2
Taking the logarithm of both sides with base 7:
z = log_7(2)
Step 4: Substitute the values into the expression
We can now substitute the values we calculated in the previous steps into the expression:
7^-(log_7(8) × log_8(7)) × 7 log_7(2)
= 7^-(x × y) × 7^z
Step 5: Simplify the powers
Using the properties of exponentiation, we can simplify the expression further.
Recall that when we multiply two powers with the same base, we add their exponents.
Therefore,
7^-(x × y) = 7^(-x × y)
And since the exponent is negative, we can rewrite it as:
7^(-x × y) = 1/(7^(x × y))
So, the expression becomes:
(1/(7^(x × y))) × 7^z
Step 6: Simplify further
Using the properties of exponentiation again, we can simplify the expression.
Recall that when we divide two powers with the same base, we subtract their exponents.
Therefore,
(1/(7^(x × y))) = 7^-(x × y)
And since the exponent is negative, we can rewrite it as:
7^-(x × y) = 1/(7^(x × y))
So, the final simplified expression is:
1/(7^(x × y)) × 7^z
That's the step-by-step breakdown of simplifying the given expression.
To simplify the given expression, let's break it down step by step:
Step 1: Evaluate log_7(8)
To find log_7(8), we need to determine what power we need to raise 7 to in order to get 8. In this case, 7^x = 8. By observation, we can see that 7^2 = 49 > 8, and 7^1 = 7 < 8. Therefore, log_7(8) will be between 1 and 2. Since we want a more precise answer, we can use logarithmic properties to rewrite 8 as 7^1.09 (approximately). This means log_7(8) ≈ 1.09.
Step 2: Evaluate log_8(7)
Similarly, to find log_8(7), we need to determine what power we need to raise 8 to in order to get 7. In this case, 8^x = 7. Using logarithmic properties, we can rewrite 7 as 8^0.88 (approximately). This means log_8(7) ≈ 0.88.
Step 3: Evaluate 7^-(log_7(8) × log_8(7))
Now that we have the values of log_7(8) and log_8(7), we can substitute them into the expression. We have 7^-(1.09 × 0.88). By simplifying the exponent, we get 7^-0.9552.
Step 4: Evaluate 7 log_7(2)
Lastly, we need to evaluate 7 log_7(2). Here, we have a logarithm with base 7. In this case, log_7(2) represents the power we need to raise 7 to in order to get 2. By observation, we can see that 7^1 = 7 < 2, and 7^2 = 49 > 2. Therefore, log_7(2) will be between 1 and 2. Since we want a more precise answer, we can use logarithmic properties to rewrite 2 as 7^0.3 (approximately). This means log_7(2) ≈ 0.3.
Step 5: Final Calculation
Now, we have 7^-(1.09 × 0.88) × 7 log_7(2) = 7^-0.9552 × 7 × 0.3. Simplifying further, we have 0.3 × 7^-0.9552. Evaluating this expression gives us the final answer.
Please note that evaluating this expression may require the use of a calculator or a computer program capable of computing complex exponentiation.