MF is Medium Frequency, generally considered to be 300kHz to 3MHz. This includes the AM Broadcast Band.
LF is Low Frequency, generally considered to be 30kHz to 300kHz (but often meaning to include even lower frequencies, VLF etc., as well).
Full size antennas (monopoles, dipoles) are obviously huge due to the long wavelengths. That's why portable AM radio use ferrite cores. Active antennas are a practical alternative for receive mode. Efficient transmit is very difficult unless you have lots of space.