For x > 1 and y = log x which one of the following is not true ?

Let $f(x)=\log x-(x-1)$.

Then, $f'(x)=\frac{1}{x}-1=\frac{1-x}{x}$

Clearly,

f'(x) < 0 for x > 1

⇒ f(x) is decreasing function for x > 1

⇒ f(x) < f(1) for x > 1

⇒ log x - (x - 1) < 0 for x > 1

⇒ log x < x - 1 for x > 1

But,

$x^2-1>x-1$ for x > 1

∴  $x^2-1>x-1$ and $\log x<x-1 \Rightarrow \log x<x^2-1$

Similarly, it can be proved that $\frac{x-1}{x}<\log x$.

Hence, (a), (b) and (d) are true. Consequently, option (c) is not true.