WheyPCRocks
Correct. Just use the next free portion of your logbook. Any "log" is sufficient just to show the required components you need to show. It could be a set of napkins if it was formal enough and identified your ratings and limitations etc. Besides it is a good talking point to instructors because of how different it is.
If you didn't have your Private in Australia and are looking to complete it in the US, most instructors will be watching you fly and can tell pretty quickly what you need to sharpen up on.
You will have to do the US based manoeuvres which differ from Oz for a checkride.
As you don't have a Private rating and are not getting certificated as a Foreign Based license nor are you doing a "conversion" , you must go through the TSA Alien training approval process.
AOPA Online: AOPA's Guide to TSA's Alien Flight Training/Citizenship Validation Rule
Additionally you will need to get a number of signoffs in your CASA book before doing the FAA checkride anyway. (Foreign based is easy to get and doesn't require this, however I have my full FAA license as well)
Off the top of my head, my first instructor needed to sign off (each was separate and following a specific phraseology that a) I can take the written/computer test, b) That I have reviewed the questions I missed on the written with the instructor, c) I meet solo privs for local and cross country, d) I meet all requirements to take the checkride (instrument, night, cross countries etc).
Remember, no night VFR, so actually do and enjoy the night flying. It is great over here.
Only other major difference is the oral portion of the checkride. But it is good to brush up on those knowledge areas. Obviously some differ from Australia, so buy the little study book for it.
Good luck!