It seems like the words expound and elaborate as a verb are snynonym which means explain something in detail.The word elucidate is also given similar meaning that explaining something clearly.The word postulate has a bit different meaning but I find also something common with the others in terms of meaning.
I'd like to ask as to how to use them properly ?
Expound means to set something forth, to explain. Elaborate means to explain something in further detail. They are quite similar, but expounding goes into detail incidentally rather than specifically.
I once rode a giraffe.
Can you expound how you did that?
This means to tell us all about it. [Edit: this might not be a very good example, as expound has the idea of an explanation of a complex idea or point of view. Since giraffe riding is probably quite simple (which is not to say easy), using the word expound in relation to it is actually a humorous exaggeration. The example in my comment, "expound my views of the relationship between the Tao Teh Ching and the New Testament," is a more typical example of when one might use the word expound.]
I once rode a giraffe.
Can you elaborate on how you did that?
This means to go into detail on the technique involved (the "on" is common but optional). So, they are indeed quite similar.
Elucidate has the specific idea of clarifying. Postulate has the idea of putting forth a theory, assertion, or argument as possibly correct. "Hypothesize" is a closer synonym than your other words. They are somewhat related, because one typically will elaborate or expound upon the idea postulated, in order to back it up.
