Determine the realtions ($\mathcal{O}$,$\Theta$,$\Omega$ ) between $f(n) = \ln(n^{c} + n^{d})$ and $g(n)=\ln(n^{a} + n^{b})$

Question

I am trying to determine the realtions ($\mathcal{O} $,$\Theta$,$\Omega$ ) between :
$$f(n) = \ln(n^{c} + n^{d})$$ $$g(n)=\ln(n^{a} + n^{b})$$ Note: $a,b,c,d>0$

I need some advice how to use the $\log$ rules to simplify it by using limits. thanks.

Answer 1

As Antonio Vargas remarked, $f(n)=\Theta(g(n))$ for any choice $a,b,c,d>0$. More precisely, $$ \lim_{n\to\infty} \frac{\ln (n^c+n^d)}{\ln(n^a+n^b) } = \frac{\max(c,d)}{\max(a,b)} $$ Indeed, the sum of two negative numbers is within the factor of two of their maximum. This factor, pulled out of logarithm, becomes a negligible additive constant.