In which of the following, ` m gt n (m,n in R)`?A. `m = (log_(2)5)^(2) and n = log_(2) 20`B. `m = log_(10)2 and n = log_(10) root(3) 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)`

In which of the following, ` m gt n (m,n in R)`?A. `m = (log_(2)5)^(2)

1.	In which of the following, ` m gt n (m,n in R)`?A. `m = (log_(2)5)^(2) and n = log_(2) 20`B. `m = log_(10)2 and n = log_(10) root(3) 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» Correct Answer - A::D (1) `m - n = (log_(2)5)^(2) - (log_(2)5+2)` ` = (log_(2)5-2)(log_(2)5+1) gt 0` ` :. m gt n ` (2) ` 2^(3) lt 10` ` or 2 lt 10^(1//3)` ` or log_(10) 2 lt log_(10)10^(1//3)` ` :. M lt n` (3) ` 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` (4) ` 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)`