There's a great deal of debate over this. People who regularly use synthetic oils swear by synthetic oils, and people who regularly use conventional oils swear by conventional oils. For most driving conditions, your engine will be properly protected by conventional oil.
However, if your vehicle is subject to extreme conditions, such as sustained high speeds, high loads, extremely dusty conditions, racing, towing, or use of a turbocharger, conventional oil may not adequately protect your engine. The key is to change your oil at the proper intervals.
Synthetic oils claim to allow a longer interval between changes. This claim is countered by those who point out that regardless of the condition of the oil, contamination from outside sources such as blow-by and moisture cannot be removed by filtering, and regardless of the lubricating properties of the oil, when contamination reaches unacceptable conditions, the oil and filter should be changed.
The main attractions of synthetic oil are its improved lubricating properties and extended oil change intervals. The trade-off is cost. Synthetic oils can cost as much as four to five times more than conventional oils. This cost increase can cancel out the savings you gain by increased change intervals.
Another factor to consider is overall maintenance. If you're going to a repair facility for your conventional oil changes every 3,000 miles, this means that a trained technician is regularly inspecting your car for potential problems. Your mechanic will check things like fluid levels, belts, hoses, tire wear patterns, under-car oil or transmission fluid leaks and other items that can help prevent breakdowns or expensive repairs. For example, replacing a weak coolant hose can prevent having that hose burst in traffic, with resultant overheating and engine damage. This type of preventive maintenance can greatly extend the life of your car. If you use a synthetic oil and only change it every 10,000 miles or more, then you're not getting your car inspected by that mechanic anywhere near as frequently, and problems may be more likely to develop between inspections.
If you're doing your own oil changes, you need to be aware of these types of inspections and perform them yourself. If you do these inspections regularly every 3,000 miles, you can possibly extend your oil change intervals by using synthetic oil; however, don't forget about problems such as moisture and blow-by contamination.
Synthetic oils do lubricate better. However, given the high quality of today's conventional oils, the high cost of synthetic oils, and the other factors involved in determining oil change intervals (and therefore cost-per-mile), there may not be an advantage in using synthetic oils for normal operation. If you're putting a lot of stress on your oil, you might consider a change to synthetic oil and staying with a short change interval. This provides the protection of synthetic oil, although your costs will increase. Those costs should be balanced against the possible cost of premature engine repairs due to excessive wear from harsh operating conditions.
There's no easy answer to the question of conventional versus synthetic oils. The answer depends on your driving, your maintenance, and your personal preferences. Generally, if your vehicle is subject to extreme conditions, synthetic oils may well be worth considering.