These are empirical models. I think their acceptance comes from nothing more than the quality of their predictions.

The log(distance) and log(frequency) components of these models have their roots in the free space propagation loss, which is the result of the inverse square law and antenna aperture for equal gain antennas at different frequencies.

And to these, the empirical models add log(transmit antenna height) and log(receive antenna height), which subtract from path loss (within certain limits of height only, and the receive antenna height term varies accoding to the environment). And then other terms to increase path loss in the presence of tall buildings, many buildings, and so on,

I think the best answer is, beyond just the free space propagation aspects, these other terms are tweaked and massaged from real-world measurements, not the other way around.

These are empirical models. I think their acceptance comes from nothing more than the quality of their predictions.

The log(distance) and log(frequency) components of these models have their roots in the free space propagation loss, which is the result of the inverse square law and antenna aperture for equal gain antennas at different frequencies.

And to these, the empirical models add log(transmit antenna height) and log(receive antenna height), which subtract from path loss (within certain limits of height only, and the receive antenna height term varies accoding to the environment). And then other terms to increase path loss in the presence of tall buildings, many buildings, and so on,

I think the best answer is, beyond just the free space propagation aspects, these other terms are tweaked and massaged from real-world measurements, not the other way around.

Thanks for you answer Mr. Albert Manfredi. I do highly agree with that. Most of wireless engineers and researchers consider the log distance law as a "theoretical fact", however, it is more close to be an empirical approximation.

Yes, but some aspects ARE just derived from theory:

Loss(dB) = 32.45 + 20logbase10(freq in MHz) + 20logbase10(dist in Km)

That's free space propagation, and can be derived (inverse square law and antenna aperture). What I'm saying is, if you go much beyond that, to accommodate antenna heights in different types of urban or rural or mountainous terrain, now you're getting to the empirical terms of the model.

The log-distance gives you a general relationship between path loss and distance that depends on the free space propagation equation. However, compared to the log-distance formula, I prefer to use the formula for log-normal shadowing. The reason is that the variation in path loss follows a Gaussian distribution. Even if the distance between the transmitter and the receiver remains the same, but the position of the receiver node changes, there can be variation in the path loss due to variation in the multi-path components. Also the gain of the transmitting antenna is not uniform. Furthermore, the effects of diffraction, scattering, and reflection might change since by changing the position of the receiver can have an effect on the LOS component. Therefore, the log-normal shadowing formula takes into account the log-distance formula plus a parameter X which is a random variable with Gaussian distribution.

Dr. Hourani's assertion that most wireless engineers regard free-space propagation as a theoretical fact could be subject to misinterpretation.  Yes, it's true that free-space attenuation follows log(distance).  However, that doesn't imply that real world propagation does.  Many of the empirically-based propagation formulas use free space as a baseline; i.e. they calculate losses relative to free space.  This does not imply in any way that engineers use free space in any way other than as a baseline.

Dr. Mahmud sums up the factors affecting land mobile propagation pretty well.  I would add one thing to his discussion.  It's true, as he says, that shadowing does cause signals with a log-normal distribution, but he neglects to mention the effects of reflections on the signal statistics.  In a completely reflective environment (i.e. no direct signal) the signal statistics will follow a combined Rayleigh/lognormal distribution, also known as a Suzuki distribution.  In an environment which also has a direct path, the signal statistics will follow a Ricean/lognormal distribution.

The log-distance path loss model is suggested by the free-space propagation theory but is not a theoretical result. However, this model:

PL(dB) = PL0(dB)  + 10 x alpha x Log10(Distance/D0)

where D0 = the reference distance (for example, D0 = 1 m for indoor environments) gives good results, with a root mean square error about 3-4 dB. For many measurement campaigns I obtained a correlation coefficient about 0.9... 0.95 between measured path loss values and  values PL(dB) computed with this model. This indicates a very good correlation. The exponent alpha > 0 can be obtained from measurements. This exponent depends on propagation environment (indoor/outdoor, urban/rural) and propagation conditions (LOS = Line of sight or NLOS = Not LOS). For a corridor it is possible to obtain alpha < 2 (LOS scenario), while alpha > 2 for NLOS environments.

You could view the log-distance model as the basis for estimating the large-scale variations of the received signal strength, as a function of distance.

The local mean value will then be represented by adding a Normal random variable thus incorporating the effects of shadowing.

You may then buid upon this model to incorporate the small scale variations due to multipath assuming no movement between the transceivers (no Doppler effect) and ideal isotropic antennas.

Small scale variations are usually modelled by the Rayleigh (for NLoS propagation), Rice (LoS propagation), Nakagami (can be used for both NLoS/LoS) or Weibull (can be used for both NLoS/LoS) distributions.

Practically, I have found Nakagami and Weibull to be more useful when fitting real RSS data, with Rice providing better results occassionaly when the measurements are collected near the sensitivity levels of the receiver (i.e. near the range limit of the receiver). 

You may also want to check out other distributions that also exist (α-μ, η-μ, or Suzuki that mixes the effects of shadowing and multipath) etc.  The log-distance model should only be used as the first step for the characterisation the wireless channel.