What is the asymptotic notation of a logarithmic function with a difference in exponents?

Question

For example:

$F(n) = \log(n) + 3\log^5(n^2) + 2\log^3(n)$

What would the asymptotic notation be since the logarithms have a different exponent. Are higher exponent logs more dominant in the notation?

Answer 1

Classify by degree:

$F(n)\sim_\infty 96\log^5(n)$.

Answer 2

Make $x=\log(n)$ to get $$F(x)=6 x^5+2 x^3+x=6x^5\left(1+\frac 1{3x^2} +\frac 1{x^4}\right)\sim 6 x^5$$