Question on log periodic dipole antenna

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.

According to literature, it's more complicated. As a simplified idea, imagine that the characteristic feed line impedance loaded by the dipole elements equals the antenna input impedance. So it's always higher than the antenna input impedance.

A good overview about LPD antenna design method and brief calculation example together with detailed references can be found in the Log-Periodic Antennas paragraph of Ballanis, Antenna Theory.