Let's say you were to to pick up a dictionary and look up a word in Chinese before the advent of any type of phonetic notation system such as Pinyin or Bopomofo. How would words in that dictionary be organized? Would you look up characters by radical? Would they be organized semantically? Did it try and account for China's massive linguistic diversity in any way?
Chinese dictionaries have arranged characters according to radicals for several centuries, then sorting on next level according to number of strokes. Works when you're looking for the meaning or pronunciation of a character you just saw. Some dictionaries and encyclopedias also attempted sorting by "theme": animals, plants, metals, etc. Of course, works only if you know a word and are looking for how it's written.
The canonical list of radicals is taken to be that used in the Kangxi dictionary, presumed to be one of the most complete dictionaries ever made on Chinese logograms:
For Japanese I can attest the problem is solved via kana: when searching the kanji for a word you know, they're ordered according to the sorting of kana (aiueo, ka ki ku...) . When you look instead for a character's meaning or its pronunciation, though, they also use the radical + number of strokes method. As a foreign learner, you're in for some surprises for some characters' official stroke steps are not what you could intuitively think of :)