The iPhone 5 requires iTunes 10.7.
iTunes 10.7 requires an Intel CPU, running Mac OS X 10.6.8 or later.
So you need to upgrade your Mac to OS X "Snow Leopard":
To upgrade from OS X 10.5.8, you must first buy Snow Leopard by calling the phone number in the Apple Online Store: 800-MY-APPLE (in the US). Snow Leopard costs $20, or $30 for a five user license. Do not obtain Snow Leopard from any source other than Apple.
Make sure your system meets Snow Leopard's requirements:
To install Snow Leopard for the first time, you must have a Mac with:
- An Intel processor
- An internal or external DVD drive, or DVD or CD Sharing
- At least 1 GB of memory (RAM) (additional RAM is recommended)
- A built-in display or a display connected to an Apple-supplied video card supported by your computer
- At least 5 GB of disk space available, or 7 GB of disk space if you install the developer tools