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Basic of Logarithms
medium
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
Solution
(b) $x = {\log _a}bc$ $ \Rightarrow $ $1 + x = {\log _a}a + {\log _a}bc = {\log _a}abc$
$\therefore {(1 + x)^{ – 1}} = {\log _{abc}}a$
$\therefore {(1 + x)^{ – 1}} + {(1 + y)^{ – 1}} + {(1 + z)^{ – 1}} = {\log _{abc}}a + {\log _{abc}}b + {\log _{abc}}c$
$ = {\log _{abc}}abc = 1$.
Standard 11
Mathematics
