The number `N=(1+2log_(3)2)/((1+log_(3)2)^(2))+(log_(6)2)^(2)` when simplified reduces to:

Options

A. A prime number
B. an irrational number
C. a real which is less than `log_(3)pi`
D. a real which is greater than `log_(7)6`

Answer

C, D

Explanation

`N=(1+2log_(3)2)/((1+2log_(3)2)^(2))+(log_(6)2)^(2)`
`log_(6)2=(1)/(log_(2)6)=(1)/(log_(2)2+log_(2)3)=(1)/(1+log_(2)3)`
let `log_(3)2=t`
`impliesN=(1+2t)/((1+t)^(2))+(1)/((1+(1)/(t))^(2))[becauselog_(2)^(3)=(1)/(log_(3)2)]`
`N=(1+2t)/((1+t)^(2))+(t^(2))/((1+t)^(2))=1`