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If a2 + 4b2 = 12ab, what is the value of log(a + 2b)?
- a)log(a/2) + log(b/2) + log 2
- b)(log a + log b – log 2) 12
- c)1/2 (log a + log b + 4 log 2)
- d)1/2(log a – log b + 4 log 2)
Correct answer is option 'C'. Can you explain this answer?
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If a2 + 4b2 = 12ab, what is the value of log(a + 2b)?a)log(a/2) + log...
Given: a2 4b2 = 12ab
To find: log(a 2b)
Solution:
We can simplify the given equation as follows:
a2 - 12ab + 4b2 = 0
This is a quadratic equation in 'a'. We can solve for 'a' using the quadratic formula:
a = [12b ± √(144b2 - 16b2)]/2
a = 6b ± 2√(5)b
Now, we can write the expression we need to find in terms of 'a' and 'b':
log(a 2b) = log[(6b + 2√(5)b) 2b]
log(a 2b) = log(6b + 2√(5)b) + log(2b)
log(a 2b) = log(2b) + log[3b + √(5)b]
log(a 2b) = log(2b) + log(b)[log(3 + √5)]
log(a 2b) = log(2b) + log(b)[1/2 log(10 + 2√5)]
log(a 2b) = log(2b) + 1/2 log(b) + 1/2 log(10 + 2√5)
Now, we can substitute for the value of log(b) using the given equation:
a2 4b2 = 12ab
log(a2 4b2) = log(12ab)
log(a2) + log(4b2) = log(12) + log(a) + log(b)
2log(a) + 2log(b) = log(12) + log(a) + 2log(b)
log(a) = 1/2 log(12) - 1/2 log(b) + log(b)
log(a) = 1/2 log(12) + 1/2 log(b)
Substituting this in the previous expression, we get:
log(a 2b) = log(2b) + 1/2 log(b) + 1/2 log(10 + 2√5)
log(a 2b) = log(2) + log(b) + 1/2 log(10 + 2√5) - 1/2 log(3)
log(a 2b) = log(2b) + log(b)[log(3 + √5)/2] + 1/2 log(10 + 2√5) - log(3)/2
log(a 2b) = log(2b)[1 + log(3 + √5)/2] + 1/2 log(10 + 2√5) - log(3)/2
log(a 2b) = log(2b)[log(3 + √5) + 1]/2 + 1/2 log(10 + 2√5) - log(3)/2
log(a 2b) = 1/2 [log(2b) + log(10 + 2√5) - log(9)]
log
To find: log(a 2b)
Solution:
We can simplify the given equation as follows:
a2 - 12ab + 4b2 = 0
This is a quadratic equation in 'a'. We can solve for 'a' using the quadratic formula:
a = [12b ± √(144b2 - 16b2)]/2
a = 6b ± 2√(5)b
Now, we can write the expression we need to find in terms of 'a' and 'b':
log(a 2b) = log[(6b + 2√(5)b) 2b]
log(a 2b) = log(6b + 2√(5)b) + log(2b)
log(a 2b) = log(2b) + log[3b + √(5)b]
log(a 2b) = log(2b) + log(b)[log(3 + √5)]
log(a 2b) = log(2b) + log(b)[1/2 log(10 + 2√5)]
log(a 2b) = log(2b) + 1/2 log(b) + 1/2 log(10 + 2√5)
Now, we can substitute for the value of log(b) using the given equation:
a2 4b2 = 12ab
log(a2 4b2) = log(12ab)
log(a2) + log(4b2) = log(12) + log(a) + log(b)
2log(a) + 2log(b) = log(12) + log(a) + 2log(b)
log(a) = 1/2 log(12) - 1/2 log(b) + log(b)
log(a) = 1/2 log(12) + 1/2 log(b)
Substituting this in the previous expression, we get:
log(a 2b) = log(2b) + 1/2 log(b) + 1/2 log(10 + 2√5)
log(a 2b) = log(2) + log(b) + 1/2 log(10 + 2√5) - 1/2 log(3)
log(a 2b) = log(2b) + log(b)[log(3 + √5)/2] + 1/2 log(10 + 2√5) - log(3)/2
log(a 2b) = log(2b)[1 + log(3 + √5)/2] + 1/2 log(10 + 2√5) - log(3)/2
log(a 2b) = log(2b)[log(3 + √5) + 1]/2 + 1/2 log(10 + 2√5) - log(3)/2
log(a 2b) = 1/2 [log(2b) + log(10 + 2√5) - log(9)]
log
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Community Answer
If a2 + 4b2 = 12ab, what is the value of log(a + 2b)?a)log(a/2) + log...
Given,
a2 + 4b2 = 12ab; adding 4ab to both sides of the equation,
we get (a + 2b)2 = 16ab
2 log (a + 2b) = 4 log 2 + log a + log b
⇒ log (a + 2b) = 1 / 2 [log a + log b + 4 log 2]
Hence, the correct option is (C).
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If a2 + 4b2 = 12ab, what is the value of log(a + 2b)?a)log(a/2) + log(b/2) + log 2b)(log a + log b – log 2) 12c)1/2 (log a + log b + 4 log 2)d)1/2(log a – log b + 4 log 2)Correct answer is option 'C'. Can you explain this answer?
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