which one of them is the dominant term and what is the big O?

Question

$3\log_8(n^2)+\log_2(n^3)$

which is the dominant term ? I think that both are dominant as it will become $6 \log_8(n) + 3\log_2(n)$..is it correct?

Answer 1

Since $\log_b(x)=\frac{\ln x}{\ln b}$ for all $b>0$, you have $$ 3\log_8(n^2)+\log_2(n^3)=\left(\frac{6}{\ln 8}+\frac{3}{\ln 2}\right)\ln n $$ None of the terms are dominant (or both).