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Thread: How do you prove this?

  1. #1
    Newbie
    Joined
    Jan 2011
    Posts
    3

    How do you prove this?

    Good Day,
    We know these facts:
    $\displaystyle x = log_a(bc)$
    $\displaystyle y = log_b(ca)$
    $\displaystyle z = log_c(ab)$

    The equation below must be proved:
    $\displaystyle x + y + z + 2 = xyz$
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  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    715
    Consider

    $\displaystyle bc = a^x ~,~ ca = b^y ~,~ ab=c^z $ we have

    $\displaystyle a^{xyz} = (bc)^{yz} = (b^y)^z (c^z)^y = (ca)^z (ab)^y = a^{z+y} b^y c^z = a^{z+y} (ca)(ab) = a^{2+z+y} bc = a^{x+y+z+2} $

    Thus , $\displaystyle xyz = x+y+z+2 $
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