Open Logic and select "New" from the "File" menu to create a blank project. In the "New Track" dialogue, select "Audio" and then select "Stereo" as its format. Click "Create," and a new audio track called "Audio 1" will appear in the arrange area. Place the audio file containing your unprocessed vocals on that track. You can drag and drop it directly from its location on your drive, or right-click on the track and select "Add Audio File." The audio file will appear as a standard audio region on the track.
Click on the audio region and then click the "Set Locators" button in the tool bar. This will loop the audio region, making it possible to adjust compression parameters without having to stop and rewind playback after each adjustment. Press "Play" and listen to the unprocessed vocals to get used to how they sound unprocessed. Compression is often quite subtle, so you will need a basis for comparison.
Click on the first available insert slot on the audio track's channel strip on the far left. From the menu that appears, select "Dynamics" and then "Compressor." This will insert, or "instantiate," Logic's built-in compressor plugin on that track. All sound from that track will now pass through this compressor. The compressor's user interface will appear.
Click on the box labelled "#default" on the compressor's interface. A menu containing various preset options will appear. Select "04 Voice," and then select "FET Vocal 01." This is one of six presets available in Logic for compressing vocals. These should not be regarded as one-click solutions for vocal compression, but rather as a good place to start. The compressor interface's controls will change to reflect the "FET Vocal 01" preset.
Press "Play" and allow the audio region to loop continually from this point on. You may want to use headphones in order to get closer to the sound. Using the "FET Vocal 01" preset should give your vocals a slight boost in volume while also making the overall sound more uniform in volume. In other words, the quieter parts of the original recording will now be slightly louder, while the louder parts will be slightly quieter.
Create a new MIDI track underneath the audio track by clicking the "+" above the track headers. Select "Software Instrument" and then click "Create." Click on the "Media" button in the toolbar, and select the "Loops" tab. Choose one of Logic's built-in drum loops, preferably one that suits the genre represented by your vocals. For our purposes, however, any drum loop will do.
Listen to the vocals and drum loop together while keeping the compressor interface visible. Turn both the "Attack" and "Release" parameters all the way to the left. The sound of the vocals will "drift" to the back of the mix and become obscured by the drums. Turn the "Attack" and "Release" knobs to the right in small increments until the vocal track becomes more prominent. Do this until it is no longer drowned out by the drum loop.
Adjust "Compressor Threshold" downwards slowly. The vocals will increase in volume as you go. Adjust this parameter until the vocals have good "presence" -- in other words, until the vocal track is at a slightly higher volume than the drums. This will bring the vocal track "forward" in the mix while keeping its volume consistent.
Experiment with different compressor settings, in particular the "Gain" and "Limiter Threshold" parameters, until you arrive at the sound you want. "Gain" increases the amount of volume available to the track, while "Limiter Threshold" limits the overall decibel level to a specific value, keeping the track's output beneath that limit at all times. Beyond the basic techniques for compressing vocals described above, compression becomes a matter of taste. When you have found the sound you want, click on "FET Vocal 01" on the compressor interface, and select "Save Setting As." Give your new vocal compression preset a name, and click "OK." Now you can use your settings in any project.