How do you prove this?

**mehdi** · February 1st 2011, 09:45 PM

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

The equation below must be proved:
$x + y + z + 2 = xyz$

**simplependulum** · February 2nd 2011, 12:04 AM

Consider

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

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

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

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