Well... Being able to 'predict' how systems and people (which are much the same really) will act in certain conditions and situations is a very usefull skill to have. 
In ancient times image you are finding your way through a landscape and you come to a void no way round (in sight) well from the logical standpoint you look at it like this 
1 how deep is the void 2 how far can I jump 3 will I make the jump based on that measurement/experience? if you didnt have logic then you would just jump and hope for the best...which is the mentality an awfull lot of people seem to employ when making decisions now...look at the sub prime crisis! Of course having too much Logic is the flip side..Imagine not leaving your house because statistics tell you that % of people get run over, shot, stabbed, attacked, abducted, struck by lightening, killed by flying debris...ect..., well logic would tell you hey dont leave the house it safer in here! so logic is like theory and then there is in practice which is usually slightly different. in essence you dont want to be too logical, or too creative, which is why most people have a proportion of both...nature intended it that way for a very good reason. 


ANSWER 

Logic:

• Enables clear thought through a rigorous demand for truthfulness;
• Because thought processes are clarified, the use of logic enables consideration of all available options for opinion and action, and decreases the persuasive power of popular opinion;
• Because available options have been fairly considered, the use of logic increases the likelihood that subsequent opinions and actions will adhere most closely to the truth.
• Adherence to the truth, while sometimes painful, is the human condition more likely to lead to satisfactory outcomes in most situations.
• Therefore, logic's relevance to everyday life is that its use improves the likelihood of satisfactory outcomes in the day-to-day decisions each person makes.

Ruecian's ANSWER 

Logic is the ability of reasoning your choices, to examine the consequence of every available action and then choose the best one. 

It has been proven that a simple rat has basic logic, 
Scientists created a machine that would create two types of sounds: 
a long beep and a short beep. They also gave the rat three small levers to pull, 
Lever 1 and lever 2 and lever 3. 
When the sceintists played a long sound or short sound, the rat would get fed a large amount of good food if it pulled the correct corosponding lever to the sound played, left lever for long beep, right lever for short beep, if the rat pulled the middle lever, it would get fed some food, but not as much, no matter what sound was played. If the rat pulled the incorect lever, it would not get fed anything for a while. The rat eventually caught on that if it pulled the correct lever it would get much better food, so it started pulling the correct levers. One day, to test the rats logic abilities, the scienctists decided to play many different sounds, long short short long long short, to see what the rat would do. 
The rat was able to think about his choices, he chose to pull the middle lever, and he took the small amount of food because he knew it was the best choice. The rat would rather not take a chance at missing out on his food. 
Without logic, everybody would act instantly on their emotions, thus making many many radical choices, without logic there would be more criminals, for example: 
you catch your girlfriend or boyfriend cheating on you, 
you have tons of choices you can make at this point 
A. some people just end the relationship and walk away 
B. some people might try to talk about it 
C. some people might even do something bad like attack somebody 

Without logic, most people would probably just attack somebody at this point because they didnt think about which choice was best, obviously either A or B is a better choice than C, but because you were angry, you didnt use logic, and just chose. Imagine if everybody chose option C for all of their choices? Life would be alot worse than it is now. 
Without logic, life would be chaos, without logic, the rat would starve >_>. 
ANI Answer Logic, is a deductive reasoning that results when the human brain calculates the most rational and acceptable outcome of any given situation and recognizes that answer as the most constructive, and consequently the most desirable. 
When faced with any situation you can divide the reactionary decision into two groups; rational or irrational, instinctual or improv, emotional or logical. 
Emotional responses tend to be the strongest desire in the moment, when followed outcomes like crimes of passion or moments of extreme passion. I tend to think that acting on an emotional paradigm feels better, and tends to be more rewarding but it also tends to precede more danger. 
Logical responses happen when the individual looks at a situation from a third party and recognizes the moral obligation or larger demographic of opinion that would rationally decide the correct decision. Why logic is so relevant to everyday life is a difficult question. I believe it stems from a natural human desire to help support a social group and thus keep everyone content. This instinctual human behavior might originate from tribal hunts in ancient times when a single person would bring back a catch and share it equally with the tribe, thus keeping the tribe alive and healthy. The center of this instinct also shares space with politics, and even our stock market of buying and sharing stock. For the record people should actually answer the question, rather than just explain what the subject is.  

http://www.answers.com/Q/Why_is_logic_relevant_to_everyday_life