I'm struggling to understand the following derivation where $n$ is a positive integer.
$$ \sum_{\ell=0}^n {n \choose \ell} 2^\ell \log 2^\ell = n \sum_{\ell=0}^{n-1} {n-1 \choose \ell} 2^{\ell+1}. $$ Splitting off the last term of the sum I can see where the $n$ factor comes from but I'm not sure why this changes the binomial coefficient.
I replace with $l$ with $k$ for better readability
$$\log(2^k)=k\cdot\log2$$
and $$k\binom nk=k\cdot\frac{n!}{(n-k)!\cdot k!}=n\binom{n-1}{k-1}$$