How does my professor go from this exponential equation to a logarithmic one?

Question

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How does the "therefore" portion work? How does that exponential equation come to equal n(lgn + 1)?

Answer 1

In the first line, $n=2^k$, so from the next to last we substitute in. Given this definition, $k=\lg n, k2^k=n \lg n$

Answer 2

HINT: $k = log_2(n) \Longrightarrow 2^k(k + 1)$

Substitute!