Basically, I've been seeing these graphs that say: Cepheid Period-Luminosity Relation and the Period axis is supposed to be "logarithmic";
How does this graph work? Since the luminosity of a star doesn't just increase, as we know, but it increases and decreases periodically, why does the magnitude keep raising?
How do I find the pulsating period of such a star through a graph like this?
This graph is not the light curve of a cepheid. It is used to find the average brightness of the star, and from that, estimate its distance.
The rate at which cepheids pulse is related to their average luminosity (and so their average absolute magnitude). This graph relates the period to their average absolute magnitude.
To use the graph, observe a cepheid for long enough to establish a period (say 30 days, for example). Then read the graph (an absolute magnitude of just under -5.5 in the example)
You now know how intrinsically bright the star is on average. By looking at its average apparent brightness, you can calculate an estimate of its distance. See How does the cepheid variables help in finding distance
