I'm trying to solve
$$ T(n) = 4T(n/4) + n \log_{10}n.$$
I'm having trouble with Iteration Method near the end. As far as I went, I obtained the General Formula as:
$$4^kT(n/4^k)+n\log n+\sum (n/4^k)\log(n/4^k)$$
And trying to get the moment when it finishes iterating:
$$(n/4^k)=1$$ $$n=4^k$$ $$\log_4n=k$$
And here I get stuck. I know I have to substitute $k$ with $\log_4n$ but after that I'm lost. Can I get a bit of help with explanation of every step?
Here are some more details:
Swapped $\log_4n$ on all $k$: $$4^{\log_4n} T(n/4^{\log_4n})+n\log_2n+\sum_{n=0}^{\log_4n}n\log n$$
From logarithm rules of $a^{log_an} = n$, it ends like this: $$(n)(1)+n\log_2n+\sum_{n=0}^{\log_4n}n\log n$$
I'm not sure how to express the sum in $n$, but as you can see already, it is $O(n\log_2n)$, and with the Master Theorem, you obtain the same result $O(n\log n)$.