Light bulbs may burn out too fast because they are screwed in too tightly. Each light bulb should be screwed in to a point at which its tip touches a tiny filament that conducts electricity, sending it into the bulb. When the bulb is screwed in too hard, this small piece of metal is bent back to a point where it does not return to its normal position. The impact is a small electrical arc that creates extreme heat and begins to melt the soldering.
Before you install a new light bulb in your kitchen exhaust fan, ensure that it corresponds to the manufacturer's recommendations for wattage. If you install a higher-watt bulb than recommended, you may cause the bulb to burn out.
If you find that your connection is fine and that you are installing the right-size bulb, check for faulty wiring to ensure that your bulb is not getting an occasional electrical surge that it cannot handle. The best way to diagnose faulty wiring is to have a qualified, licensed electrician look at your wiring system.
Turn off the power before examining any wiring or connections in light bulb sockets. The tiny filament the light bulb connects to can shock you if you touch it when the power is on. If you find that your problem is a connection, help ensure the longevity of future bulbs by screwing them in slowly and with the power on so you know the exact moment the light comes on. You needn't screw the bulb in much farther than that, just another quarter-turn to make sure it's in far enough. When working with the power on, wear rubber gloves to protect against shock.