Minimization of $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$? I am trying to find the minimal value of the expression: 
$log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$
I think experience gives that the variables should be equal, if so then the minimal value is 6, but this not true in general.
Any hints or help will be greatly appreciated. 

You can rewrite this as
$$
\left(x+\frac{1}{x}\right)+\left(y+\frac{1}{y}\right)+\left(z+\frac{1}{z}\right)
$$
where $x=\log_{a}b$, $y=\log_{b}c$, and $z=\log_{c}a$.  (These three variables are not independent: there is the additional constraint that $xyz=1$.)  Each of the three identical terms has a minimum value of $2$, which it assumes at $x=1$ (resp. $y=1$ or $z=1$).  So the whole expression can't be smaller than $6$, which can be achieved by simply letting $a=b=c$.
