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Find the value(s) of x satisfying the equation 16 log4 (log3x) = log3X - (log3X) - (log3X)2 + 1
- a)1/√3
- b)6
- c)3
- d)both (A) and (C)
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Find the value(s) of x satisfying the equation 16 log4 (log3x) = log3X...
Let log3 x = a
⇒ 42log4 a = a - a2 + 1 ⇒ 4log4a2 = a - a2 + 1
⇒ a2 = a - a2 + 1
⇒ (a - 1) (a + 1/2) = 0
⇒ a = 1 or a = -1/2
⇒ log3 x = 1 or log3 = -1/2
log3 x > 0 [since log4(log3 x) term is present]
so x = 3
⇒ 42log4 a = a - a2 + 1 ⇒ 4log4a2 = a - a2 + 1
⇒ a2 = a - a2 + 1
⇒ (a - 1) (a + 1/2) = 0
⇒ a = 1 or a = -1/2
⇒ log3 x = 1 or log3 = -1/2
log3 x > 0 [since log4(log3 x) term is present]
so x = 3
Most Upvoted Answer
Find the value(s) of x satisfying the equation 16 log4 (log3x) = log3X...
To solve the equation, let's simplify it step by step:
First, let's simplify the right side of the equation:
log3X - (log3X) - (log3X)^2
= log3X - log3X - (log3X)(log3X)
= - (log3X)(log3X)
Now, let's rewrite the equation using the simplified form:
16 log4 (log3x) = - (log3X)(log3X)
Next, let's use the change of base formula on the left side of the equation:
log4 (log3x) = log(log3x) / log(4)
Now, let's rewrite the equation again using the changed base:
16 log(log3x) / log(4) = - (log3X)(log3X)
Now, let's simplify the left side of the equation:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's cross-multiply to eliminate the fraction:
16 log(log3x) = - log3X * log3X * log(4)
Now, let's simplify the right side of the equation:
16 log(log3x) = - (log3X)^2 * log(4)
Now, let's divide both sides of the equation by log(4):
16 log(log3x) / log(4) = - (log3X)^2
Now, let's simplify the left side of the equation:
16 log(log3x) / log(4) = - (log3X)^2
Now, let's rewrite the equation again using the changed base:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's simplify the left side of the equation:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's cross-multiply to eliminate the fraction:
16 log(log3x) = - log3X * log3X * log(4)
Now, let's simplify the right side of the equation:
16 log(log3x) = - (log3X)^2 * log(4)
Now, let's divide both sides of the equation by log(4):
16 log(log3x) / log(4) = - (log3X)^2
Now, let's simplify the left side of the equation:
16 log(log3x) / log(4) = - (log3X)^2
Now, let's rewrite the equation again using the changed base:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's simplify the left side of the equation:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's cross-multiply to eliminate the fraction:
16 log(log3x) = - log3X * log3X * log(4)
Now, let's simplify the right side of the equation:
16 log(log3x) = - (log3X)^2 * log(4)
Now, let's divide both sides of the equation by log(4):
16 log
First, let's simplify the right side of the equation:
log3X - (log3X) - (log3X)^2
= log3X - log3X - (log3X)(log3X)
= - (log3X)(log3X)
Now, let's rewrite the equation using the simplified form:
16 log4 (log3x) = - (log3X)(log3X)
Next, let's use the change of base formula on the left side of the equation:
log4 (log3x) = log(log3x) / log(4)
Now, let's rewrite the equation again using the changed base:
16 log(log3x) / log(4) = - (log3X)(log3X)
Now, let's simplify the left side of the equation:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's cross-multiply to eliminate the fraction:
16 log(log3x) = - log3X * log3X * log(4)
Now, let's simplify the right side of the equation:
16 log(log3x) = - (log3X)^2 * log(4)
Now, let's divide both sides of the equation by log(4):
16 log(log3x) / log(4) = - (log3X)^2
Now, let's simplify the left side of the equation:
16 log(log3x) / log(4) = - (log3X)^2
Now, let's rewrite the equation again using the changed base:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's simplify the left side of the equation:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's cross-multiply to eliminate the fraction:
16 log(log3x) = - log3X * log3X * log(4)
Now, let's simplify the right side of the equation:
16 log(log3x) = - (log3X)^2 * log(4)
Now, let's divide both sides of the equation by log(4):
16 log(log3x) / log(4) = - (log3X)^2
Now, let's simplify the left side of the equation:
16 log(log3x) / log(4) = - (log3X)^2
Now, let's rewrite the equation again using the changed base:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's simplify the left side of the equation:
log(log3x) / log(4) = - (log3X)(log3X) / 16
Now, let's cross-multiply to eliminate the fraction:
16 log(log3x) = - log3X * log3X * log(4)
Now, let's simplify the right side of the equation:
16 log(log3x) = - (log3X)^2 * log(4)
Now, let's divide both sides of the equation by log(4):
16 log
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Community Answer
Find the value(s) of x satisfying the equation 16 log4 (log3x) = log3X...
Let log3 x = a
⇒ 42log4 a = a - a2 + 1 ⇒ 4log4a2 = a - a2 + 1
⇒ a2 = a - a2 + 1
⇒ (a - 1) (a + 1/2) = 0
⇒ a = 1 or a = -1/2
⇒ log3 x = 1 or log3 = -1/2
log3 x > 0 [since log4(log3 x) term is present]
so x = 3
⇒ 42log4 a = a - a2 + 1 ⇒ 4log4a2 = a - a2 + 1
⇒ a2 = a - a2 + 1
⇒ (a - 1) (a + 1/2) = 0
⇒ a = 1 or a = -1/2
⇒ log3 x = 1 or log3 = -1/2
log3 x > 0 [since log4(log3 x) term is present]
so x = 3
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