If-x-log-a-bc-y-log-b-ca-and-z-log-c-ab-then-prove-that-x-y-z-xyz-2-

Question Number 196582 by MATHEMATICSAM last updated on 27/Aug/23

If x = \log_{a} b c, y = \log_{b} c a and z = \log_{c} a b

then prove that x + y + z = x y z - 2 .

Answered by som(math1967) last updated on 27/Aug/23

x = {l o g}_{a} b c \Rightarrow 1 + x = {l o g}_{a} a + {l o g}_{a} b c

\Rightarrow 1 + x = {l o g}_{a} a b c \Rightarrow \frac{1}{1 + x} = {l o g}_{a b c} a

s a m e w a y \frac{1}{1 + y} = {l o g}_{a b c} b

\frac{1}{1 + z} = {l o g}_{a b c} c

∴ \frac{1}{1 + x} + \frac{1}{1 + y} + \frac{1}{1 + z} = {l o g}_{a b c} a b c = 1

\Rightarrow \frac{1 + y + 1 + x}{(1 + x) (1 + y)} = \frac{z}{1 + z}

\Rightarrow 2 z + y z + z x + 2 + x + y = z (1 + x + y + x y)

\Rightarrow 2 z + y z + z x + 2 + x + y

= z + z x + y z + x y z

∴ x + y + z = x y z - 2