Question 1045915
{{{x = log(15,2) = log(15,30) - log(15,15) = log(15,30) - 1}}}.
Similarly, {{{y = log(10,3) = log(10,30) - log(10,10) = log(10,30) - 1}}}, and
{{{z = log(6,5) = log(6,30) - log(6,6) = log(6,30) - 1}}}.

===> {{{xy+yz+zx = (log(15,30) - 1)( log(10,30) - 1)+( log(10,30) - 1)(log(6,30) - 1)+(log(6,30) - 1)(log(15,30) - 1)}}}

= {{{-2log(10,30)-2log(15,30)-2log(6,30)+log(15,30)*log(10,30)+log(10,30)*log(6,30)+log(6,30)*log(15,30) + 3}}}.


Also, 

{{{2xyz = 2(log(15,30) - 1)( log(10,30) - 1)(log(6,30) - 1)}}}


= {{{2log(15,30)*log(10,30)*log(6,30) - 2log(15,30)*log(6,30)-2log(10,30)*log(6,30)+2log(6,30) - 2log(15,30)*log(10,30)+2log(15,30)+2log(10,30) - 2}}}


===> {{{xy+yz+zx+2xyz = 2log(15,30)*log(10,30)*log(6,30) - log(15,30)*log(6,30) - log(10,30)*log(6,30) - log(15,30)*log(10,30) +1}}}


= {{{2/(log(30,15)*log(30,10)*log(30,6)) - log(30,15)/(log(30,15)*log(30,10)*log(30,6)) - log(30,10)/(log(30,15)*log(30,10)*log(30,6)) - log(30,6)/(log(30,15)*log(30,10)*log(30,6)) + 1}}}

= {{{(1/ (log(30,15)*log(30,10)*log(30,6)) )(2-log(30,15) - log(30,10) - log(3,6)) +1}}}

={{{(1/ (log(30,15)*log(30,10)*log(30,6)) )(2-log(30,900)) +1}}}

= {{{(1/ (log(30,15)*log(30,10)*log(30,6)) )(2-2) +1 = highlight(1)}}}