Why A log-periodic array uses elements whose lengths and spacings scale by a constant geometric ratio, so different portions of the array become the active radiating region at different frequencies. That makes the feedpoint impedance and pattern stay usable over a very broad frequency range, often covering several ham bands in one antenna. The trade-off is that only a few elements are active at any one frequency, so gain is modest for the number of elements used.
Watch out The choice about gain per element is the opposite of the truth: in a Yagi all elements work together on one narrow band, so a Yagi gives more gain per element than a log-periodic. Harmonic suppression is also backwards, since a broadband antenna radiates harmonics readily rather than rejecting them.
Log-periodic = Lots ofPtterns... simpler: LP means 'Lots of bandwidth, Puny gain per element.'