Log4 4(7/17) - log4 3(21/86) + log4 48
We can break this down using the properties of logarithms:
log4 4(7/17) = log4 (7/17) because loga a = 1 for any base a
log4 3(21/86) = log4 3 + log4 (21/86) because loga (b*c) = loga b + loga c
log4 48 = log4 (4*12) = log4 4 + log4 12 = 1 + log4 12 because loga (b*c) = loga b + loga c and loga a = 1
Now we can substitute and simplify:
log4 (7/17) - log4 3 - log4 (21/86) + 1 + log4 12
= log4 [(7/17)/(3*(21/86)*4))] + log4 12 because loga b - loga c = loga (b/c)
= log4 [7/(3*17*21*4/86)] + log4 12
= log4 [7/(3*17*21/43)] + log4 12
= log4 [7*43/(3*17*21)] + log4 12
= log4 (1/3) + log4 12^1/2 because 7*43 = 301 and 21*3*17 = 1071, which have a common factor of 3
= log4 (12^1/2 / 3) + 1/2
= log4 (2) + 1/2 because 12^1/2 = sqrt(12) = 2*sqrt(3)
= 1/2 + log4 2
Therefore, log4 4(7/17) - log4 3(21/86) + log4 48 simplifies to 1/2 + log4 2.
3
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To solve this expression, we can start by using the properties of logarithms.
The first step is to simplify each logarithm separately and then combine them. Let's break it down step by step:
1. Log4 4(7/17):
- Since the base of the logarithm is 4, we can rewrite it in exponential form as 4^x = 4(7/17).
- Now we can solve for x: 4^x = 4(7/17).
- Simplifying further, we get: x = 7/17.
- Therefore, Log4 4(7/17) simplifies to 7/17.
2. Log4 3(21/86):
- Similar to the previous step, we rewrite it in exponential form as 4^x = 3(21/86).
- Solving for x, we get: x ≈ 0.5745.
- Therefore, Log4 3(21/86) simplifies to approximately 0.5745.
Now, let's combine both simplified logarithms:
3. Log4 48:
- We can rewrite it in exponential form as 4^x = 48.
- To find x, we calculate: x ≈ 3.
- Therefore, Log4 48 simplifies to 3.
Now, we can substitute the simplified logarithms back into the original expression:
Log4 4(7/17) - log4 3(21/86) + log4 48
= 7/17 - 0.5745 + 3
= 10.0315 + 3
≈ 13.0315
Therefore, the value of the expression Log4 4(7/17) - log4 3(21/86) + log4 48 is approximately 13.0315.