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Sunday, May 10, 2015

2015/047) Prove that if x = log_a(bc),y = log_b(ca),z = log_c(ab) then the value of xyz - x - y - z is 2


we have
x= log_a(bc)

so 1 + x = 1 + log_a(bc) = log_a(a) + log_a(bc) = log_a(abc)

or \dfrac{1}{1+x} = log_{abc}(a) \cdots (1)

similarly
\dfrac{1}{1+y} = log_{abc}(b) \cdots (2)
\dfrac{1}{1+z} = log_{abc}(c) \cdots (3)
 
 Adding (1), (2) and (3) we get
\dfrac{1}{1+x}+ \dfrac{1}{1+y}+ \dfrac{1}{1+z} = log_{abc}(a) + log_{abc}(b) + log_{abc}(c) = log_{abc}(abc) = 1

or
(1+y)(1+z) + (1+z)(1+x) + (1+x)(1+y) = (1+x)(1+y)(1+ z)

or

1 + yz + y + z + 1 + xz + x + z + 1 + xy + x + z = 1 + x + y + z + xy + yz + zx + xyz

or 2 + x + y+ z = xyz

or xyz – x – y -z = 2

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