7 ^ (- (log_7(8) * log_8(7))) * 7 * log_7(2)
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)
To simplify the expression 7 ^ (- (log_7(8) * log_8(7))) * 7 * log_7(2), we can break it down step-by-step.
Step 1: Simplify log_7(8):
We know that log_7(8) is the exponent to which we raise 7 to get 8. Therefore, log_7(8) = 1.
Step 2: Simplify log_8(7):
Similarly, log_8(7) is the exponent to which we raise 8 to get 7. This value is irrational and cannot be expressed easily. We can use a calculator to find an approximate value, which is approximately 0.876.
Step 3: Multiply log_7(8) and log_8(7):
log_7(8) * log_8(7) = 1 * 0.876 = 0.876.
Step 4: Evaluate the exponent:
Now we can substitute the value we found back into the original expression:
7 ^ (-0.876) * 7 * log_7(2)
Step 5: Simplify further:
7 ^ (-0.876) is equivalent to 1 / 7 ^ 0.876. 7 ^ 0.876 can be calculated using a calculator, and it is approximately 2.863.
After substituting this value back into the expression, we get:
1 / 2.863 * 7 * log_7(2)
Step 6: Evaluate log_7(2):
log_7(2) is the exponent to which we raise 7 to get 2. Therefore, log_7(2) = 1/2.
Substituting this value back into the expression, we get:
1 / 2.863 * 7 * 1/2
Step 7: Simplify the expression:
1 / 2.863 is equivalent to approximately 0.349.
Substituting this value back into the expression, we get:
0.349 * 7 * 1/2
Step 8: Evaluate the multiplication:
0.349 * 7 * 1/2 = 1.2225.
Therefore, the simplified expression is approximately 1.2225.
To solve this expression, we need to simplify it step by step. Let's break it down:
Step 1: Determine the values of logarithms.
- log_7(8): This represents the logarithm of 8 to the base 7. Since 7^1 = 7 and 7^2 = 49, we know that log_7(8) is between 1 and 2. We'll use an approximation of log_7(8) ≈ 1.167.
- log_8(7): This represents the logarithm of 7 to the base 8. Similar to the previous step, we can determine that log_8(7) is less than 1 because 8^1 = 8 is larger than 7. We'll use an approximation of log_8(7) ≈ 0.879.
Step 2: Calculate the exponent in the expression.
- Multiply log_7(8) by log_8(7): 1.167 * 0.879 ≈ 1.026.
Step 3: Calculate the base expression.
- Raise 7 to the power of the negative exponent from Step 2: 7^(-1.026) ≈ 0.615.
Step 4: Multiply with the remaining factors.
- Multiply the result from Step 3 by 7: 0.615 * 7 = 4.305.
Step 5: Multiply with the last factor.
- Multiply the result from Step 4 by log_7(2): 4.305 * log_7(2).
Since log_7(2) is not easily simplified to a decimal value, we'll express the final result as an expression: 4.305 * log_7(2).
Therefore, the simplified expression is 4.305 * log_7(2).