A log-periodic dipole array (LPDA, or just "log periodic") is a beam made of many dipoles on a boom, each longer than the one in front, all connected to the same feed line. It trades some gain for bandwidth: one antenna, one coax and a good SWR across a whole range of frequencies, such as several HF bands.
How it works
Every element is the same ratio longer than its neighbor, and so is every gap. The working part slides along the boom with frequency.
Every dipole is a resonant half-wave antenna at some frequency. Because the lengths run in a geometric series, there is always a short stretch of boom where elements are close to a half wavelength at the signal frequency. That active region does the radiating, and it slides along the boom as you change frequency. Longer elements behind it act as a reflector and shorter ones in front as directors, so the beam points toward the short end.
Both the lengths and the spacings change by a constant ratio, so the electrical behaviour repeats each time the frequency is multiplied by a fixed amount. Plotted on a logarithmic frequency axis it repeats periodically: hence the name. The exam describes this as element length and spacing varying logarithmically along the boom.
Gain for bandwidth
Schematic, not measured. The log-periodic gives up peak gain to cover a wide frequency range.
A Yagi uses every element for one narrow band, so it has more gain per element. In a log-periodic most elements are idle at any one frequency, so the gain per element is lower, but the pattern and the feed point impedance stay steady over a wide range. Wide bandwidth is the advantage; extra gain is not.
In practice
Multiband HF versions let one rotatable antenna cover, for example, 20 through 10 m. They are long, heavy and wind-loaded, because the longest element must suit the lowest frequency.
At VHF and UHF the same idea is a familiar sight as a television-style array, and is used for wideband monitoring and testing.
Expect a front-to-back ratio typically in the mid-teens of dB or better, depending on the design.
Outside the design range the gain and SWR fall away, so check that every band you want sits inside it.