In which of the following, ` m gt n (m,n in R)`?

Options

A. `m (log_(2)5)^(2)` and `n = log_(2)20`
B. `m =log_(10)2` and `n = log_(10)3sqrt(10)`
C. `m = log_(10)5. log_(10)20` and `n = 1`
D. `m = log_(1//2)((1)/(3))` and `n = log_(1//3)((1)/(2))`

Answer

A, D

Explanation

(A) `m - n = (log_(2)5)^(2) - (log_(2)5 + 2)`
`= (log_(2)5 - 2)(log_(2)5 + 1) gt 0`
`:. m gt n`
(B) `2^(3) lt 10`
or `2 lt 10^(1//3)`
or `log_(10)2 lt log_(10)10^(1//3)`
`:. m lt n`
(C) `m = log_(10)5.log_(10)20`
`= (log_(10)10 - log_(10)2)(log_(10)10 + log_(10)2)`
`=(1 - log_(10)2)(1 + log_(10)2)`
`=1 - (log_(10)2)^(2)`
`lt 1`
`:. m lt n`
(D) `m = log_(1//2) ((1)/(3)) = log_(2)3` and `n = log_(1//3)((1)/(2)) = log_(3)2`
`:. (m)/(n) = (log_(2)3)/(log_(3)2) = (log_(2)3)^(2) gt 1`
`:. m gt n` (as `n gt 0)`