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Question

If x=1+logabc,y=1+logbca,z=1+logcab, prove that xy+yz+zx=xyz

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Solution

x=1+logabc
=logaa+logabc
x=loga(abc)1x=1loga(abc)=logabc(a)
Similarly,
y=logb(abc)1y=logabc(b)
z=logc(abc)1z=logabc(c)
Now, add all 3
1x+1y+1z=logabc(a)+logabc(b)+logabc(c)
=logabc(abc)
1x+1y+1z=1
xy+yz+zx=xyz

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