The set of values of x for which `"log"_(e) x gt (x-2)/(x)`, is

Options

A. `(1, oo)`
B. `(1, 2)`
C. R
D. `(2, oo)`

Answer

D

Explanation

`"Let "phi (x) = "log"_(e) x - (x-2)/(2)`. Then,
`phi' (x) = (1)/(x) - (2)/(x^(2)) = (x-2)/(x^(2)) gt 0, "if" x gt 2`
` rArr phi (x) " is increasing on " (2, oo)`
`rArr phi (x) gt phi(2) " for all" x in (2, oo)`
`rArr phi(x) gt 0 " for all" x in (2, oo)`
`rArr "log"_(e) x gt (x-2)/(x) " for all" x in (2, oo)`