I am looking at an example in my algorithms textbook, and I cannot seem to figure out how the following simplification occurred (my logarithm skills are a bit rusty):
From this: $$\sqrt{n}c\sqrt{n}\lg\lg\sqrt{n}+n\le{cn\lg\lg n}$$
To this: $$nc(\lg\lg n-\lg2)+n\le{cn\lg\lg n}$$
And finally to this: $$-clg2+1\le{0}$$
I am mainly confused by the second step. I don't follow all the steps that lead to getting $\lg\lg n - \lg2$
Any help in clarifying this is greatly appreciated.
$$\log\left(\log n^{1/2}\right)=\log\left(\frac{1}{2}\log n\right)=\log\frac{1}{2}+\log\log n=-\log 2+\log\log n$$