More than just logarithm...

Algebra Level 2

If l o g b a ⋅ l o g c a + l o g c b ⋅ l o g a b + l o g a c ⋅ l o g b c = 3 log_{b}a \cdot log_{c}a + log_{c}b \cdot log_{a}b + log_{a}c \cdot log_{b}c = 3 , where a ≠ b ≠ c a \neq b \neq c , a , b , c > 0 a,b,c > 0 , and a , b , c ≠ 1 a,b,c \neq 1 , then evaluate a b c abc


The answer is 1.

Shabarish Ch by Shabarish Ch , 22, India

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1 solution

Nivedit Jain
Feb 1, 2017

Easy one. Let loga=x Logb=y Logc=z This can be written as x^2/yz + y^2/xz + z^2/xy=3 Which is x^3 + y^3 + z^3 = 3xyz 2 sol possible X=y=z first but a,b,c are not equal. So we have 1 option left X+y+z=0 So Loga + logb + logc =0 Logac=. - logb Logac= log(1/b) Comparing ac=1/b So ABC =1 This is ourreq answer.

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