Sorry, David and Mario, but tbe circle of fifths has nothing to do with being
'logical' or 'following'. I did study music theory in college.
The circle of fifths is merely a chord progression in
which each of the chords has a certain mathematical relation to
the preceding chord. Each chord is exactly five semitones tones higher than
the previous chord. You can use it for different purposes, but the only
requirement for a chord progression to be a circle of fifths is that the
chords be in a certain mathematical relation to each other.

In other words, let's say that you play an A chord. Now it does not matter
if it is an A major or minor or 7th or 11th chord or a diminished chord. If you
want to use a circle of fifths chord progression the next chord you play
would be a D chord (again it doesn't matter if it is major minor 9th etc.),
which is 5 semitones up from an A chord.

The next chord you play would be 5 semitones up, a G chord. Then another 5 semitones up would be a C chord. So, if you keep doing this 12 times you will be going from A to D to G to C to F to A# to D# to G# to C# to F# to B to E to A. So, in 12 steps, by using the circle of fifths, you will end up right where you started (it does not matter what chord you started on) which is why it is called a 'circle', because you eventually come back, after 12 chords, to the same chord you started on.

Now, if you are only using 3 chords that are in a circle of fiths progression, let's say G to C to F, then that does not really qualify as a circle of fiths progression. You really need to use at least 4 chords, such as G to C to F to A#, to qualify as an official circle of fifths progression, although that is not a hard and fast rule.

The importance of the circle of fifths progression is that it can make your music sound more interesting. The reason it is making your music sound more interesting is because you are frequently modulating to a different key. For example, if you are in the key of G and you play a C chord
you are still in the key of G. But if you then play a F chord you are now either in the key of F or the key of C, depending on which chords you play next. And if you keep playing even more chords in the circle of fifths you will continue to modulate to different keys and this will keep your listening audience alert and intrigued.

Of course, if you are interested in writing top 40 style songs, or songs for film, tv, or ads, you should never use the circle of fifths, because, in those styles of music, they are just looking for musically boring songs that sound similar to songs the listeners have heard before. Things were alot different back in the 60s and 70s when there was alot more musical experimentation, in the top 40, than in the past few decades.

To sum up, the circle of fifths is merely a chord progression in which every chord is 5 semitones higher than the previous chord and it can be used anywhere in a song, verse, chorus or bridge, and it will make your song sound more interesting and more musically sophisticated.

But do not be hesitant about using the circle of fifths just because it is based on a
mathematical pattern. After all, all music is different variations of mathematical patterns. There have even been some computer programs that allow you to input some melodies by a certain songwriter you like and then the computer will create some melodies in the style of that songwriter. These programs are not yet for sale, they have only been research projects so far But it won't be too long before these programs are for sale and you can bet that the major labels will want to ensure that the records by their artists sound enough like their previous hits, and the labels will use this sort of program to do so.

Matador is beautiful,a symphony of style
Excitement is ecstatic, passion places bets
Gracefully he bows to ovations that he gets
But the hands that are applauding are slippery with sweat
And saliva is falling from their smiles

Phil Ochs- Crucifixion