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If a, b and c are distinct positive numbers not equal to 1 and if (logca)(logba) + (logab)(logcb) + (logbc)(logac) = 3, then the value of abc is
  • a)
    0
  • b)
    1
  • c)
    2
  • d)
    3
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
If a, b and c are distinct positive numbers not equal to 1 and if(logc...
Given (logca)(logba) + (logab)(logcb) + (logbc)(logac) = 3

Or, (loga)3 + (logb)3 + (logc)3 = 3(loga)(logb)(logc)
Or, log a + log b + log c = 0
(We know that a3 + b3 + c3 = 3abc when a + b + c = 0)
Thus, log abc = 0 or, abc = 1
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Most Upvoted Answer
If a, b and c are distinct positive numbers not equal to 1 and if(logc...
To solve this problem, we can use the property of logarithms that says if log(a) + log(b) = log(c), then a * b = c.

Let's start by manipulating the given expression:
(log(ca) * log(ba)) * (log(ab) * log(cb)) * (log(bc) * log(ac)) = 3

Applying the property of logarithms, we can rewrite this expression as:
log(ca * ba) * log(ab * cb) * log(bc * ac) = 3

Now, let's simplify each logarithmic expression:
log(c * b * a) * log(a * b * c) * log(b * c * a) = 3

Since the logarithm of a product is equal to the sum of the logarithms of the factors, we have:
log(abc) + log(abc) + log(abc) = 3

Simplifying further:
3 * log(abc) = 3

Dividing both sides by 3:
log(abc) = 1

Now, using the definition of logarithms, we can rewrite this equation as:
10^1 = abc

Simplifying:
abc = 10

However, we are given that a, b, and c are distinct positive numbers not equal to 1. Therefore, the only possible value for abc is 10, which is not one of the given options.

Hence, the given answer key is incorrect. There is no correct option among the provided choices (a, b, c, or d).
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If a, b and c are distinct positive numbers not equal to 1 and if(logca)(logba) + (logab)(logcb) + (logbc)(logac) = 3, then the value of abc isa)0b)1c)2d)3Correct answer is option 'B'. Can you explain this answer?
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