In absence of a tester, I'd take the cover off a nearby outlet and look at the color of the attached wires. If the outlet is wired correctly, the wire that's attached to the smaller, vertical blade is the hot wire.

You may want to pick up a tester like this which can distinguish the hot lead from the neutral (e.g., on an electrical outlet).

All right. Here's the wiring diagram. You are correct that the tan wire is actually white. And, my big mistake was not realizing one of the fixture wires was still hiding inside.

When a person works too long without a break, the fixture wires are too short, and there's only room for a few fingers to do the twisting, well, it can make for a stressed out diy-er.

So, my question now is

Can I join the blue and black fixture wires together with the house hot wire so that one switch can turn on both the light and fan at the same time? I did try it and only the light works. Have I not got a good connection with the fan wire or will this just not work?

And, thank you Ron and Lawrence. I appreciate your help.

Of course you can control the fan and light with one switch but, of course, the light and fan each need the hot and neutral wires connected to them.

I tried to upload a photo of the wiring diagram.

On the fixture, there is only 1 neutral wire and of course 2 hot wires.

I suppose that I should take apart the 3 hot wires and retry. Maybe they are not touching well.

I tested the fan & ligh separately, so I know they each work.