First off, answer a question if you could:

Illustrator uses vectors, yes? So, does that mean any picture made in illustrator has the ability
to change size without losing resolution? Because I tried making it in photoshop and it's terrible if
I try to make it bigger.

Second off, I need this logo made in illustrator or any program meant to make logos that'll change
size and resolution often:

Just the book render!

I only have 40 forum gold to give, so whoever can do it for that amount?

you can do that with ps without getting a shitty resolution. photoshop has a path tool as well

I don't know how to do that either lol.

But Illustrator is all resolution-independent, so it's still better?

Or is that what you mean by path