Path loss = 20 log (4 * pi * r / lambda)
[Note: use log base 10]
Path loss = signal attenuation. Unit: dB
pi = 22 / 7
r = distance between transmitter and receiver. Unit: m
lambda = wavelength of signal. Unit: m
Wavelength = C/f
C = speed of electromagnetic waves in free space
= 299792458. Unit: m / s
f = frequency of signal. Unit: Hz
Let r = R * (2 ^ x)
Path loss = 20 log (4 * pi * R * L / lambda)
= 20 log (4 * pi * R / lambda) + 20 log (2 ^ x)
= 20 log (4 * pi * R / lambda) + 6x
The above derivation implies range(R) doubles every 6dB of path loss.
The path loss is +8.519dB more over a given range for the 2.4 GHz compared to 900MHz for the same range. In other words operating at 900 MHz exhibits a significantly longer range than is possible at 2.4 GHz.