ref. www.salsburg.com/Log-Periodic.pdf (?)

I can't answer your question, but some thoughts...

Obviously the units of length and diameter don't matter since they're cancelled out. The "120" is probably the same 120 as the exact value of the Zo of free space (120*Pi = 377 ohms).  The "2.25" looks like it might be a 'conglomeration' of several other constants.

The formula is obviously incomplete (simplified) because it doesn't include the spacing between the ends of the dipole elements at the feedpoint.  Imagine if the dipole elements were of very large diamter and brought very close together - they could form an easily adjustable (spacing) capacitor across the feedpoint.  Obviously you could then make the element lengths slightly longer or shorter and have some ability to tune your antenna to different feedpoint impedances.

Thanks much for the comments. The salsburg.com ref is the Peter Rhodes, K4EWG, write up on the log periodic from the ARRL Ant. HB of 1974. I suspect that much of it was based on the work of Carrel and Isbell, listed in the bibliography.

I think the expression is most like the one for the reactance of a dipole: j120*ln(L/D)-1]/tan(pi*L/lambda).

Simplified though the expression may be, it seems to work. For example, if I model lpda designs using numbers from the nomogram for Ro,SWR, tau given in the Rhodes reference you mention, it works for Ro values between 60 and 90 ohms.

The reason for the original question is that I was wondering if I could rely on it for making a pcb based lpda.
My feeling is that the dipole impedance term would have to be modified by a velocity factor that was a mix of air and pcb dialectric. However, given that the impedance of a dipole in an array is found by solving a complex matrix consisting of self and mutual impedances, I guess I'm just amazed the expression works at all.

Maybe if I could figure out exactly what the original expression was intended to represent, I could substitute the correct value into the expression in the Rhodes reference for phase line impedance. Since the latter expression is a function of tau, sigma, Ro and dipole impedance, I'm guessing I could use it for pcb designs if I could get the dipole term right - so it would help if I knew where it came form or how it was derived. I have some early Carrel and Isbell papers on order. Maybe I'll find an answer in one of them.  

Check out "Antenna Enginerring Handbook" by Henry Jasik first edtion chapter 18, section 3 (pages 18-10 thru 18-32)for some insight.  
Note: See page 14-3 in the second edtion version of the same book :"Antenna Enginerring Handbook" by Richard Johnson & Henry Jasik.