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Question

Let x=1+logabc, y=1+logbac and z=1+logcab then prove that xy+yz+zx=xyz.

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Solution

Given x=1+logabc.......(1),
y=1+logbac.......(2)
and z=1+logcab......(3).

From (1) we have,

x=logaa+logabc

or, x=logaabc(loga+logb+logc=log(abc)).

Similarly from (2) and (3) we get, y=logbabc and z=logcabc.

Now,

1x+1y+1z=1logaabc+1logbabc+1logcabc

=logabca+logabcb+logabcc(logab=1logba)

=logabcabc

1x+1y+1z=1

So,

1x+1y+1y=1

or, xy+yz+zx=xyz.

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