3,000-5,000 miles change your oil!! There is nothing wrong with synthetic oils. My suggestion for you would be to use something for higher mileage vehicles. They have an additive that protects against friction for older or higher mileage vehicles.
Here’s what to do: Check your owner’s manual and follow the manufacturer’s recommendation. If it says change the oil every three months or 3,000 miles and you haven’t traveled 3,000 miles in that time, change it anyway. But, I think you’ll find that the recommendations are much longer than that. For instance, Acura suggests that I change the oil in my car every 7,500 miles or one year.
Even if you only drive 3,000 miles a year the lubrication/detergent properties of your engine oil will break down. For my customers I recommend every 3,000 miles or 6 months. I have 2 vehicles that get driven no more than 2,000 miles a year, this is the schedule I follow.
On the same note with the milage your car accumilates over 1 year I suspect you do mostly very short drives. you should take your vehicle out for a fair drive (at least 15 miles) at least once a month or so to get the operating components and fluids up to temperature. Short trips that do not allow your car to warm up to normal operating temperature tend to cause condensation (of water) in the crankcase.
I’ll agree with hobbabob - it depends on climate and if you make short trips often or long trips rarely. In good conditions and infrequant, long trips, you can get away with longer change intervals.