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) 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

A, D

Explanation

(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)`