Main Difference
The main difference between Logic and Rational is that the Logic is a study of inference and demonstration and Rational is a quality or state of being agreeable to reason.

Logic
Logic (from the Ancient Greek: λογική, translit. logikḗ), originally meaning “the word” or “what is spoken”, but coming to mean “thought” or “reason”, is generally held to consist of the systematic study of the form of valid inference. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. (In ordinary discourse, inferences may be signified by words like therefore, hence, ergo and so on.)
There is no universal agreement as to the exact scope and subject matter of logic (see § Rival conceptions, below), but it has traditionally included the classification of arguments, the systematic exposition of the ‘logical form’ common to all valid arguments, the study of inference, including fallacies, and the study of semantics, including paradoxes. Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid19th century), and recently logic has been studied in computer science, linguistics, psychology, and other fields.

Rational
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason. Rationality implies the conformity of one’s beliefs with one’s reasons to believe, and of one’s actions with one’s reasons for action. “Rationality” has different specialized meanings in philosophy, economics, sociology, psychology, evolutionary biology, game theory and political science.
To determine what behavior is the most rational, one needs to make several key assumptions, and also needs a logical formulation of the problem. When the goal or problem involves making a decision, rationality factors in all information that is available (e.g. complete or incomplete knowledge). Collectively, the formulation and background assumptions are the model within which rationality applies. Rationality is relative: if one accepts a model in which benefitting oneself is optimal, then rationality is equated with behavior that is selfinterested to the point of being selfish; whereas if one accepts a model in which benefiting the group is optimal, then purely selfish behavior is deemed irrational. It is thus meaningless to assert rationality without also specifying the background model assumptions describing how the problem is framed and formulated.

Logic (adjective)
logical

Logic (noun)
A method of human thought that involves thinking in a linear, stepbystep manner about how a problem can be solved. Logic is the basis of many principles including the scientific method.

Logic (noun)
The study of the principles and criteria of valid inference and demonstration.

Logic (noun)
The mathematical study of relationships between rigorously defined concepts and of mathematical proof of statements.

Logic (noun)
A modeltheoretic semantics.

Logic (noun)
Any system of thought, whether rigorous and productive or not, especially one associated with a particular person.
“It’s hard to work out his system of logic.”

Logic (noun)
The part of a system (usually electronic) that performs the boolean logic operations, short for logic gates or logic circuit.
“Fred is designing the logic for the new controller.”

Logic (verb)
To engage in excessive or inappropriate application of logic.

Logic (verb)
To apply logical reasoning to.

Logic (verb)
To overcome by logical argument.

Rational (adjective)
Capable of reasoning.
“Man is a rational creature.”

Rational (adjective)
Logically sound; not contradictory or otherwise absurd.
“His statements were quite rational.”

Rational (adjective)
Healthy or balanced intellectually; exhibiting reasonableness.
“rational conduct”

Rational (adjective)
Of a number, capable of being expressed as the ratio of two integers.
“¾ is a rational number, but √2 is an irrational number.”

Rational (adjective)
Of an algebraic expression, capable of being expressed as the ratio of two polynomials.

Rational (adjective)
Expressing the type, structure, relations, and reactions of a compound; graphic; said of formulae.

Rational (adjective)
Expressing a physical object.
“A rational table is physical, a written table is neither.”

Rational (noun)
A rational number: a number that can be expressed as the quotient of two integers.
“The quotient of two rationals is again a rational.”

Rational (noun)
The breastplate worn by Israelite high priests.
“1609, DouayRheims Bible, Exodus 28:15”
“And thou shalt make the rational of judgment with embroidered work of divers colours, according to the workmanship of the ephod, of gold, violet, and purple, and scarlet twice dyed, and fine twisted linen.”