6 dB = 10^(6/10) ~= 4X the power, where the square root of 4X equals a
doubling the range (assuming an omnidirectional antenna)?

I'm hoping I can extrapolate from the above statement to calculate the dB
gain for the $150 Linksys WRT300N router which claims a 4X range
improvement (so I can compare the $50 antenna's effect with that of the
802.11n router).

Following your lead, the power improvement necessary for a 4X range
improvement is 4^2 = 16X power gain.

This 16X power gain then equates to about 12 dB (since 12 dB = 10^[12/10]
~= 16X power).

So, is it safe to calculate that the claimed 4X range improvement of the
Linksys WRT300N wireless broadband router can be compared to that of a 12
dB gain omnidirectional antenna?

Beverly