Unfortunately, it is not possible to use PicoLog with the PicoScope 4424. PicoLog is unlikely to be developed further as PicoScope6's logging capabilities are likely to be developed further.
You could use PicoScope to capture your waveforms over a period of time and save them to a file - there is an alarm feature that you can use to automatically save the current or all buffers to a file. You could adjust the timebase as required and set the maximum number of waveforms to be stored in a buffer. How long are you looking to capture data for?