If x = logc (ab), y = loga (bc), z = logb (ca), then which of the following is…
2023
If x = logc (ab), y = loga (bc), z = logb (ca), then which of the following is correct?
- A.
xyz = 1
- B.
x+y+z=1
- C.
(1 + x)-1 + (1 + y)-1 + (1 + z)-1 = 1
- D.
(1 + x)-2 + (1 +y)-2 + (1 + z)-2 = 1
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Correct answer: C
Given: x = log_c(ab), y = log_a(bc), z = log_b(ca).
Express each variable using common logarithms: x = (log a + log b)/log c, y = (log b + log c)/log a, z = (log c + log a)/log b.
Compute 1 + x: 1 + x = 1 + log_c(ab) = (log c + log(ab))/log c = log(abc)/log c. Therefore 1/(1 + x) = log c / log(abc).
Similarly, 1/(1 + y) = log a / log(abc) and 1/(1 + z) = log b / log(abc).
Add the three fractions: 1/(1 + x) + 1/(1 + y) + 1/(1 + z) = (log c + log a + log b)/log(abc) = log(abc)/log(abc) = 1.
Conclusion: (1 + x)^{-1} + (1 + y)^{-1} + (1 + z)^{-1} = 1, which verifies the correct statement.