Expert: Ahmed Salami - 9/19/2010

Question
If x=log(12)6,y=log(18)12,z=1og(24)18

Prove That
1+xyz = 2yz
Terms in the bracket are bases of log terms  

Answer
Hi Dhananjay,
x = log(12) 6
implies 12^x = 6
taking natural logarithms of both sides
ln 12^x = ln 6
x(ln 12) = ln 6
x = ln 6 / ln 12
Similarly, y = log(18) 12 and z = log(24) 18 result in
y = ln 12 / ln 18
and
z = ln 18 / ln 24
Now,
xyz = (ln 6 / ln 12)(ln 12 / ln 18)(ln 18 / ln 24)
= ln 6 / ln 24
1 + xyz = 1 + (ln 6 / ln 24)
= (ln 24 + ln 6) / ln 24
= ln 24.6 / ln 24
= ln 144 / ln 24

yz = (ln 12 / ln 18)(ln 18 / ln 24)
= ln 12 / ln 24
2yz = 2.ln 12 / ln 24
= ln 12^2 / ln 24
= ln 144 / ln 24

and hence the proof.

Regards