For `x gt 1` and y = lod x which one of the following is not true ?

Options

A. `x-1gt y`
B. `x^2-1gty`
C. `ygtx-1`
D. `(x-1)/xlty`

Answer

C

Explanation

Let f(x) = logx-(x-1).
Then `f'(x)=1/x-1=(1-x)/x`
Clearly ,
`f'(x) lt of or x gt 1. `
`rArr`f(x) is decreasing function for `x gt 1`
`rArr f(x) lt f(1) for x gt 1`
`rArr logx-(x-1) lt 0 for x gt 1`
`rArr log x lt x-1 for x gt 1.`
But
`x^2-x-1 for x gt 1`
`therefore x^2-1 gt x-1 and log x lt x-1 rArr log x lt x^2-1`
Similarly, it can be proved that `(x-1)/x lt logx` Hence (a) , (b) and (d) are true . Consequently , option (c) is not true