Grade 1210 grade maths
If x=1+logabc, y=1+logbac, z=1+logcab, Prove that xy+yz+zx = xyzanswer it fassstttt
If x=1+logabc, y=1+logbac, z=1+logcab, Prove that xy+yz+zx = xyz
answer it fassstttt
1 Answer
Vikas TU
Dear student
x = 1 + loga (bc)
= loga (a) + loga (bc)
= loga (abc)
or, 1/x = log(abc) a .....(1)
similarly, y = 1 + logb (ca) = logb (abc)
or, 1/y = log(abc) b ......(2)
and z = 1 + logc (ab) = logc (abc)
or, 1/z = log(abc) c ......(3)
now, 1/x + 1/y + 1/z = log(abc) a + log(abc) b + log(abc) c
= log(abc) (a × b × c)
= log(abc) (abc)
= 1
or, 1/x + 1/y + 1/z = 1
or, (yz + yz + xy)/xyz = 1
or, xyz = xy + yz + zx
hence, xyz = xy + yz + zx
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