Question

If $x = \log_{a}(bc),y = \log_{b}(ca),z = \log_{c}(ab),$ then which of the following is equal to 1

• A. $x + y + z$
• B. $(1 + x)^{- 1} + (1 + y)^{- 1} + (1 + z)^{- 1}$
• C. $xyz$
• D. None of these

Answer 1

Answer: (B)

$(1 + x)^{- 1} + (1 + y)^{- 1} + (1 + z)^{- 1}$

Answer 2

$x = \log_{a}bc \Rightarrow 1 + x = \log_{a}a + \log_{a}bc = \log_{a}abc$

$\therefore(1 + x)^{- 1} = \log_{abc}a$

∴$(1 + x)^{- 1} + (1 + y)^{- 1} + (1 + z)^{- 1} = \log_{abc}a + \log_{abc}b + \log_{abc}c$

$= \log_{abc}abc = 1$.