I recommend getting a little intuition about the behavior of the functions , and by plotting their graphs. You can do this, for example, in Excel or online at Function Grapher Online (just found this site). There is also a List of information graphics software in Wikipedia.

The goal of this is not to lower the standards of rigor and to give proofs in terms of one picture. Rather, it helps to appreciate to what extend some functions dwarf others when arguments become large.

A brief answer is that (I assume that all numbers are positive) will beat for any . In turn will beat for all and . When the function arguments become sufficiently large, the dominant function would tell you, for example, whether you are near Mars or near Jupiter while a lower-order function will describe the location of your friend on Earth who went to the nearest grocery to by milk.