log(3,6)*log(9,6)+log(4,6)*log(18,6)

$$log_63*log_69+log_64*log_618\\\\
=log_63*2log_63+log_64*log_6(9*2)\\\\
=log_63*2log_63+log_64*(log_69+log_62)\\\\
=log_63*2log_63+2log_62*(2log_63+log_62)\\\\
=log_63*2log_63+4log_62log_63+2(log_62)^2\\\\
=2(log_63)^2+4log_62log_63+2(log_62)^2\\\\
=2[(log_63)^2+2log_62log_63+(log_62)^2]\\\\
=2[(log_63+log_62)^2]\\\\
=2[(log_66)^2]\\\\
=2[1^2]\\\\
=2\\$$

log(3,6)*log(9,6)+log(4,6)*log(18,6) =

log(3,6)*[log(3,6) + log(3,6)] + log(4,6)*[log(3,6) + log(3,6) + log(2,6)]

log(3,6) = ~0.613, log(4,6) = ~0.774, & log(2,6) = ~0.387.

So, 0.613*[0.613 + 0.613] + 0.774*[0.613 + 0.613 + 0.387]

= 0.752 + 1.248 

= 2.

Sorry, for the re-post of Aziz's excellent answer....I was confused by the notation in this question and I wanted to get some "practice" in to make sure I understood it !!!......I'll give Aziz full credit, here!!!

-------------------------------------------------------------------------------------------------------------------------

log(3)/log(6) = 0.6131471927654584

log(9)/log(6) = 1.2262943855309169

So

[log(3)/log(6)]*[log(9)/log(6)] = 0.75189895999232446469782360080696

And

log(4)/log(6) = 0.7737056144690832

log(18)/log(6) = 1.6131471927654584

So

[log(4)/log(6)]*[log(18)/log(6)] = 1.24810104000767559661689567573888

So we have

0.75189895999232446469782360080696 +1.24810104000767559661689567573888 =

2.00000000000000006131471927654584 ≈  2

Very nice, Melody....that is well laid out !!!!

3 points from me!!!

Thanks Chris.