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How do I properly connect a backlit mirror to an existing light switch?

I feel like I'm asking an absolute newbie question here, but: is it possible to plug the lighted bathroom mirrors into an existing outlet so I can use the bathroom light switch to turn on the bathroom mirror?

I have an outlet here (like a ceiling light socket, N/L wires and ground). However, the bathroom mirror that was really intended came with a power plug to plug into the outlet. Then I would only be able to switch the lights on the mirror itself. That's not what I was looking for.

Unfortunately, I rarely find instructions online on how to plug in a proper backlit mirror, and I'm just wondering if the outlet as designed is only for an individual lamp, or if I can/should somehow "convert" the 3-wire outlet to achieve my goal.

• bathroom
• light