(Claimed simple) reasoning with logarithms that I cannot follow. Suppose $k \leq {c \times \log n}/{\log \log n}$ where $c$ is a constant.
Then we have the following reasoning:
$\log k^k = k \log k \leq (c \log n/\log \log n)(\log c + \log \log n)\leq (c+1)\log n$
The two steps here that seem like magic are:
$k \log k \leq (c \log n/\log \log n)(\log c + \log \log n)$
and 
$(c \log n/\log \log n)\times(\log c + \log \log n)\leq (c+1)\log n$
Can anyone break them down please?

The underlying meaning is the following: The quantity $k=k(n)$ depends on $n$, and we have an estimate of the form
$$k(n)\leq{c\log n\over\log\log n}\qquad(n\to\infty)\ .$$
Then for sufficiently large $n$ we can say that
$$\log k^k\leq(c+1)\log n\ .$$
Proof.  We may write
$$\eqalign{\log k^k&=k\log k\leq{c\log n\over\log\log n}(\log c+\log\log n-\log\log\log n)\cr
&\leq c\log n\left({\log c\over\log\log n}+1\right)\leq c\log n\left({1\over c}+1\right)\qquad(n>n_0)\ .\cr}$$
