Log4 (4 (7/17)) - Log4 (3(21/68)) + log4 (48)
AAAaannndd the bot gets it wrong yet again!
2 + log4 (64/81)
= 2 + log4(64) - 4log4(3)
= 2+3 - 4log4(3)
= 5 - 4log4(3)
My apologies for the mistake in my previous response. You are correct, the correct answer is 5 - 4log4(3). Thank you for bringing this to my attention and for providing the correct solution.
To simplify the given expression, we can use the properties of logarithms. Specifically, we can use the following rules:
1. logₐ(b) - logₐ(c) = logₐ(b/c)
2. logₐ(b^c) = c * logₐ(b)
Let's break down the given expression step by step:
1. log₄(4(7/17)):
- We can simplify the logarithm using rule #2. Since log₄(4) = 1, we can replace it with 1.
- log₄(1 * (7/17))
- log₄(7/17)
2. log₄(3(21/68)):
- Similarly, we replace log₄(3) with 1.
- log₄(1 * (21/68))
- log₄(21/68)
3. log₄(48):
- We can simplify log₄(48) using rule #2. We need to find the exponent that 4 must be raised to in order to get 48.
- 4^3 = 64 (since 4 * 4 * 4 = 64)
- So, log₄(48) = 3
Now, by substituting the simplified expressions back into the original expression:
log₄(7/17) - log₄(21/68) + log₄(48)
We can now calculate the individual logarithmic values:
log₄(7/17) ≈ -0.612
log₄(21/68) ≈ -1.184
log₄(48) = 3
Finally, substituting these values into the expression:
-0.612 - (-1.184) + 3
= -0.612 + 1.184 + 3
≈ 3.572
Therefore, the final result is approximately 3.572.
To solve this problem, we can use the properties of logarithms to simplify the expressions and combine any like terms.
First, we can simplify the logarithms that involve division:
log4 (7/17) = log4 7 - log4 17
log4 (3(21/68)) = log4 3 + log4 (21/68) = log4 3 + (log4 21 - log4 68)
Now we can substitute these simplified forms back into the original expression:
Log4 (4 (7/17)) - Log4 (3(21/68)) + log4 (48)
= Log4 (4/17) - (log4 3 + (log4 21 - log4 68)) + log4 48
Next, we can use the logarithmic property of multiplication to simplify the first term:
Log4 (4/17) = log4 4 - log4 17 = 2 - log4 17
Substituting this into the expression gives:
2 - log4 17 - (log4 3 + (log4 21 - log4 68)) + log4 48
We can use the logarithmic property of addition to combine the terms inside the parentheses:
2 - log4 17 - log4 3 - log4 21 + log4 68 + log4 48
Finally, we can combine the remaining terms using the logarithmic properties of addition and subtraction:
2 + log4 (68*48/17*3*21)
= 2 + log4 (3264/3591)
= 2 + log4 (576/633)
= 2 + log4 (64/81)
= 2 + log4 ((4/3)^(-4))
= 2 - 4
= -2
Therefore, the answer is -2.