I often find two numbers roughly "in the same ballpark" if they are within a factor of about $e$ of each other. For example, if I know computers generally cost upward of $\$1000$, then $\$2700$ would probably be the most I would a priori feel is a reasonable upper limit for computer prices.
Is this just a coincidence, or is there some mathematical deep reason why $e$ is inherently the "best" order-of-magnitude exponent?
I think that we subconsciously think in a logarithmic way and since the root of 10 is around 3.162, we think of 3 as half way the order of magnitude. So anything under 3,000 we perceive in the order of magnitude of 1,000 and anything over that we perceive as being in the 10,000 order of magnitude.
In other words if 1,000 is 3 in 10 based logarithm and 10,000 is 4 in 10 based logarithm then 3,162 is the 3.5 on this scale and anything under 3,000 is closer to 3 and almost everything over 3,000 is closer to 4.