first, the main formula of logarithms:
Log (a/b)= log a - logb
loga{a/b}=loga{a} - loga{b}
loga{a}=1
the simplification is as follow:
A log3(3/4) +log3(4/5)+log3(5/6)+log3(6/7)+log3(7/8)+log3(8/9)
let ---- = ------------------------------------------------------------------------------------
B log(4/5) +log(5/6) +log(6/7) +log(7/8) +log(8/9) +log(9/10)
and A = log3(3) -log3(4) + log3(4) -log3(5)+ log3(5)-log3(6)+log3(6)-log3(7)+log3(7)-log3(8)+log3(8)-log3(9)= log3(3)-log3(9) after simplification
and log3(3)=1, log3(9)=log3(3)^2=2log3(3)=2x1
finally A = 1 - 2 = -1
with a same method we can find B= log4 - log10= log(4/10)=log(2/5)
finally A/B= -1 / log(2/5)