How do you determine the amount of shares of common stock is issued and outstanding with 500000 shares authorized?

The 500,000 shares authorized means nothing, other than it won't be more than 500,000.

It depends on what information you've been given. Financials for real companies say how many shares they have. For homework problems, they can give you anything and make you figure it out - there isn't any "one" way.

However, a common way textbooks do it is by giving you the total dollar value worth at par, and then giving par per share. Keep in mind, if you have par, the stock has to be listed at par. You'd get the dollar value by doing:
par x shares = total dollar value.

If that's what they've given you, it's just a matter of taking that info and working it backwards. If that info isn't available, we'd have to know what is available.