Big O Proof for Logarithmic Function
$$5\log_2(2n)+7=5\log_2 n+5\log_2 2+7=5\log_2 n +12$$
Since for example $12\le 12\log_2 n$ for $n\ge 2$, it follows that
$$5\log_2 n +12\le 5\log_2 n +12\log_2 n=17\log_2 n $$
for $n\ge 2$. This yields $O(\log n) $.