No exporting -
Moving the iPhoto library is safe and simple - quit iPhoto and drag the iPhoto library intact as a single entity to the external drive - depress the option key and launch iPhoto using the "select library" option to point to the new location on the external drive - fully test it and then trash the old library on the internal drive (test one more time prior to emptying the trash)
And be sure that the External drive is formatted Mac OS extended (journaled) (iPhoto does not work with drives with other formats) and that it is always available prior to launching iPhoto
And backup soon and often - having your iPhoto library on an external drive is not a backup and if you are using Time Machine you need to check and be sure that TM is backing up your external drive
But when photos get exported, the info changes too. The time & date for the photos becomes the the time that the back-up was created. How do I keep the info the same as the ones on iphoto?
There are two kinds of metadata involved when you consider jpeg or other image file.
One is the file data. This is what the Finder shows. This tells you nothing about the contents of the file, just the File itself.
The problem with File metadata is that it can easily change as the file is moved from place to place or exported, e-mailed, uploaded etc.
Photographs have also got both Exif and IPTC metadata. The date and time that your camera snapped the Photograph is recorded in the Exif metadata. Regardless if what the file date says, this is the actual time recorded by the camera.
Photo applications like iPhoto, Aperture, Lightroom, Picasa, Photoshop etc get their date and time from the Exif metadata.
When you export from iPhoto to the Finder new file is created containing your Photo (and its Exif). The File date is - quite accurately - reported as the date of Export. However, the Photo Date doesn't change.
The problem is that the Finder doesn't work with Exif.
So, your photo has the correct date, and so does the file, but they are different things. To sort on the Photo date you'll need to use a photo app.