Big-O proof of inclusion

Question

I'm working on this proof of inclusion:$$\log_2(8^n)\in{\mathcal O(n)}$$ $$\log_28^n-cn\leq0$$ for all $n>n_0$. Is there a log rule that I can use to further simplify before I plug random values to check the statement?

Answer 1

You should check if: $$\lim\limits_{n\to \infty}\frac{\log_2 8^n}{n}=C\geqslant 0$$

You have from $\log$ ruls:

$$\require{cancel}\lim\limits_{n\to \infty}\frac{\cancel n \log_2 8}{\cancel n}=\log_2 8=\color{red}3$$

Therefore $\log_2 (8^n) \in \mathcal O(n)$

BTW, the rule is $\log m^n=n \log m$

Answer 2

$\log_a (b^n) = n\log_a(b)$ for all $a,b,n$.