Synch iPhone 5s with MacBook 10.5.8.

Read the 5s box. It will say what version of iTunes the phone requires. Very likely whatever is current at the time of purchase of the 5s. That means you will likely have to upgrade to Lion at least (OSX 10.7 -- see the system requirements for iTunes in the iTunes download tab at the top of this page).

System requirements for OS X Lion and Mac OS X v10.6 - http://support.apple.com/kb/HT4949

Note that upgrading to 10.7 will result in no longer being able to use PPC generation applications. IT may also require upgrading RAM in the computer. Apple has minimums but they really mean it and most people go with double minumum.

Unfortunately that tends to happen at Apple Stores when you start mentioning anything older than 4 years. They simply are not familiar with anything that is not capable of running Apple's 'latest and greatest'. (*cough cough*). Hunt down the oldest assistant in the Store and he/she may vaguely remember computers running Leopard but for most of them it is inconceivable anybody would still be doing do and it doesn't occur to them anybody would when buying newer gadgets. To be honest, you'll get better answers here when it comes to using "vintage and obsolete" equipment. 🙂

We bought a 5s a month ago. If I recall correctly it wasn't even labeled on the box and when I asked in the Apple Store the guy didn't even know what iTunes version would be required. My wife has a 4 month old MacBook Pro with Mavericks and iTunes 11 so I didn't know if we would have to upgrade iTunes (and still don't because she never syncs any media to her work phone so we haven't plugged it in). Maybe if you can find out for a very specific phone in the store you might get away with using iTunes 11 which I believe only requires Snow Leopard (OSX 10.6) which would definitely be possible on any Intel Mac like your MacBook. Snow Leopard does still support PPC generation software.