Logos is a subset of philosophy, and science is derived from it in some distinct ways, starting with the concepts of rationality established by Greek philosophers.

The idea that God must be rational, ergo the world must have rational order, was developed by men who were exposed the greek philosophies later on, when technology had advanced a bit further.

there was what one might call "rudimentary" science done even by the ancient greeks, but when one discusses modern science in terms of the scientific method and the standards of observation, it doesn't actually show up until....not going give a date, can't remember right now, I think it's sometime near the middle of the second millenia but don't quote me.

so, do they rely on each other? yes.

Science is by far the greater method for determining truth about reality, though logic combined with scientific knowledge allows us understand things without directly observing them, through things like necessity, or theorize about them, such as the formation of hypothesis and their concurrent falsification conditions.

for example, when we form the hypothesis "sunlight helps plants grow", and we set up an experiment, we have a control that allows us falsify it, and the structure is inherently logical.

IF the only difference between two plants is the amount of sunlight, AND the one in the sun grows faster, THEN sunlight increases plant growth. That is the experiment you would derive from the condition, and it's truth is determined by the concept of logical necessity developed by philosophy. Both A and B being true necessitate the conclusion, there is no other logical choice. Science did not come up with that concept, and while it is a core concept of science, it is taken directly from logic.