will open the last library that it used when it is started up.

When you launched iTunes with the external drive turned off, iTunes could not find the last library and so it opened another one.
It should have given you some dialog box that it was doing so.

Shut down iTunes, make sure the external is on and ready, and then launch iTunes with the Option Key held down (Mac) or the Shift Key (Windows). You did feel it necessary to share what equipment you were using or what version of what OS or the version of iTunes.

When the dialog box appears asking for the location of your iTunes library, point iTunes to the library on the external drive.
All should now now be working correctly.

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