If you see both the foreground and the moon clearly and sharp, it's either a double exposure or stacked photos. The moon is moving quickly enough that you want at least 250th exposure to capture it clearly. If you expose for the foreground the moon will be a bright blur.
Also, that moon that looks so big on rising really isn't. That's an optical illusion that your camera won't capture. Again two exposures are necessary to make a photo that looks like what you saw. Back in the day, I'd shoot one landscape shot, switch lenses to a 200mm and re-set at 250/f8, repositioning the frame so the moon ended up in the right part of the frame. Now it's easy with Photoshop and layers. Just place the sharp moon over the blurry spot that you got with setting for the foreground exposure. You'll probably have to adjust the size of the moon so it looks right.
Here's a shot without the second exposure:
michael mckee
My Port Townsend – A City in Photographs