A log-periodic antenna is basically a balanced transmission lines with radiators attached to it. I would guess the transmission line Z is around 70ohm like other 1/4 wavelength dipole antennas.
If that is the case in your design you would simply use a transmission line calculator for balanced lines and plug in 70ohm and teh distance between the feed lines.
The distance and the diameter of the lines is what creates a specific impedance so if you change on of these parameters you must adjust the other to keep the same specific impedance.