Why A log-periodic dipole array is built from a series of dipole elements whose lengths and spacings scale by a constant ratio as you move along the boom, so the dimensions grow logarithmically. Each element is resonant near a different frequency, and the active region shifts along the boom as you change frequency. That geometry is what gives the LPDA usable gain and SWR over a very wide frequency range, unlike a Yagi that is tuned to one band.
Watch out The choices about impedance, gain or SWR varying periodically or logarithmically with frequency invert the idea: it is the physical element dimensions that follow the logarithmic progression, and the result is that performance stays roughly constant across the design range.
The 'log' in log-periodic is in the boom, not the performance: lengths and spacings scale logarithmically.