How does $\log(x^2 + 1)$ become $\log(2x^2)$? My textbook attempts to take the big O of $\log(x^2 +1)$. It proceeds by saying $x^2 + 1 \le 2x^2$ when $x \ge 1$. But I don't know how it came up with this idea.
Question:
Why set $x^2+1$ to a random value to be $2x^2$? Why $2$ of all numbers? Why not $x^2$ or $x^3$? 

You could have taken $3 x^2$ if you wanted, or $x^3$.  You'd end up with the same result.
But not $x^2$: it's certainly not true that $x^2 + 1 \le x^2$.
