With l'hopital's rule I can show that $f(x)=2^x\log{(1+2^{-x})}$ goes to $1$ as $x$ goes to $\infty$.

What I intuitively don't get though is why apparently $2^x$ and $\log{(1+2^{-x})}$ exactly balance each other such that the limit to $\infty$ is $1$.

For example, if the base of the first term would be slightly higher, then the limit would go to $\infty$.

Any hints for the intuition of this result?