How are these two expressions equal/the same?

Question

The expressions:

I don't understand where the sqrt root goes and how we go from 1/2 to 1/4.

Answer 1

Because $$\log(x^a)=a\log(x)$$

So: $$\log(\sqrt{x^2+1})=\log[(x^2+1)^\frac12]=\frac12\log(x^2+1)$$

Answer 2

You can use that $$\ln(x^r)=r\ln(x)$$ for $$x>0$$ and $$r\neq 0$$