Is all of mathematics based on logic? I do not understand the discipline in mathematics that is called "mathematical logic". For me, all of mathematics is based on logic and that is what makes it the exact science. Every theorem or lemma or result should be based on correct reasoning, and there should be some true logic behind it. Or am I mistaken?

Are there some results that cannot be treated with logic? What is the difference between the discipline of "mathematical logic" and the logic used in mathematics? Are there known results in mathematics that are not based on logic?

For example: 


  Find $x$ in $(1)$: $$(1)\quad x+1=0.$$ 
  The logic is to find $x=-1$ by subtracting $-1$ and no one else can say otherwise.


For me, this is mathematics. Even though a lot of theorems are hard to understand (at least for me), there must be some kind of logic behind them. Am I right?
Generally speaking, mathematical logic investigates the nature and limitations of logical systems. For example, consider the question: "can every true statement be proved?" This is a problem about logic. When we study geometry, algebra  and other branches of mathematics we are interested in using logical thinking, but the purpose is to understand something about shapes or symmetries.
I think the same way as you think as if we see, mathematics all trick until now has been concerned with "=" , ">" , "&lt;" , etc have a meaning and that is for = it means that what is written on left side is exact same as what is on the right side and so on. So we'll Yes actually, mathematics is completely logical as you can not reach a statement that is true by just moving numbers here and there but you can get multiple forms of a true statement by displacing numbers and that's what math is. Understanding multiple forms of the same truth. For example : 
If we write :

X-2 = 0 

Then it means that any number in which we subtract 2 happens to be 0 then exactly we can also say that 

X = 2

Because 2 is such a number .

And both of these statements refer to the same number , 2 , hence we can see clearly that by twisting numbers, we just see the same truth other way.