Question

在由正整数组成的等比数列{an}中，若a4a5a6=3,log3a1+log3a2+log3a8+log3a9的值为

Answer 1

等比数列{an}中

a4*a5*a6=a5*(a4*a6)=a5*a5^2=a5^3=3 [^表示指数，a5^2=a4*a6]

于是log3(a5^3)=1=3log3(a5)

于是log3(a5)=1/3

而a1*a2*a8*a9=a1*a9*a2*a8=a5^2*a5^2=a5^4

[a5是a1和a9的中项，又是a2和a8的中项]

=根据对数性质

log3a1+log3a2+log3a8+log3a9

=log3(a1*a2*a8*a9) [对数相加，底数不变，真数相乘]

=log3(a1*a9*a2*a8)

=log3(a5^4)

=4log3(a5)

=4/3