Let $c$ be an exponential growth rate, and $a(n)$ any expression in $n$ (sequence, polynom, function,...).
Consider $$ \log(2^{cn}+a(n)). $$ I am asking myself what properties (increasing, decreasing, etc.) must be fullfilled for $a(n)$ in order to have $$ \lim_{n\to\infty}\frac{\log(2^{cn}+a(n))}{\log(2^{cn})}=1, $$ (notation for that: $\log(2^{cn}+a(n))\sim\log(2^{cn})$.
Isn't is necessary that $a(n)$ grows slower than $2^{cn}$ or equal as $2^{cn}$, i.e. $a(n)\in O(2^{cn})$?