How do you find log of a Z* number?

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I understand modular arithmetic on a base level.

I was wondering how can one find log(base 5)20 in Z * base23.

I'm very confused about what Z* represents as I've been trying to figure it out for hours. Is it just a mod of a number? How would you compute it by hand? It is very hard to find any information on that. If you can possible give a simple example with a whole number, because I have no idea how you can calculate log in Z*.