Although I know that I'm going to get slammed by atheists, I feel that I need to point out that there are places in our English translations that use the words for "divorce" and "put away" interchangeably.
And yes, context is important. There is a difference of context when I tell my wife that I love her than when I tell my mother the same thing. I can't see how there can be any argument with that.
Anyway, learning about the words from the original Greek text is a part of study and maturing. Divorcing and putting away are two different things.
Of course, divorce is not pleasing to God. But it is a means of escaping a destructive relationship, where both can remarry. Putting away is throwing the person out on the street while remaining married. This person commits adultery if they go to someone else because they're still married.
If you study about the woman at the well, Jesus recognized all of her husbands as being her husbands. He said the man she currently had was not.
Even though we are to avoid sin, we also need to use the bible as its best commentary. Men's traditions were a lot of why Jesus rebuked the Pharisees so harshly. The same things can still happen today.
There may be many who disagree. There may be atheists that say I'm contradicting myself. However, if a person truly studies this out, the logic is there.