A useful rule of thumb for bath vanities is to use a minimum of two 60-watt incandescent bulbs, which typically produce approximately 800 lumens each for a total of 1,600 lumens. In your application you have several choices:
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You could choose a fixture with standard medium-base sockets and install LED bulbs that fit the sockets. You would need LED bulbs equivalent to the standard 60-watt incandescent bulbs with approximately 800 lumens each.
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Considering there is only one fixture in your bathroom, you could go to a higher lumen output, and perhaps go to a 75-watt equivalent and use a dimmer, which would allow you to lower or increase the light level as needed.
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You could use an integrated LED fixture, which does not have light bulbs but instead is integrated with the LED lighting as a permanent part of the fixture. LEDs have extremely long life (about 50 times that of incandescent) making this essentially a lifetime fixture purchase. With this option you would want a fixture in the range of 1200-2000 lumens. Integrated LED fixtures are more efficient in delivering light, making it possible to go with a slightly lower lumen output (1200 lumens).
To look at the many options available, visit an ALA-member showroom where a certified lighting consultant can show you a variety of design options.