Who wins when we compare polynomials and logarithmic functions? Look at a picture.
Eventually, after not too long, the polynomial will pull ahead of the logarithmic function. This makes sense, because the polynomial is curved upwards, while the logarithmic curve looks like it's flattening out.
The logarithmic curve never flattens out; it has no horizontal asymptotes, and grows without bound, but it does so slowly.
If we take some limits, we find
and
Think of these three types of functions as if they are racing. Power functions are like powerful race horses; polynomials (Polly want a cracker?) are like parrots fluttering along; and logarithmic functions are like logs, plodding and slow. The category a function belongs to is determined by its leading term, that is, the individual term that "grows the fastest."