Many theologians accept the criticism that their beliefs lack rational proof, and still insist that they are right to believe in the existence of God. However, it is much harder for them to be confronted with the problem of evil, because this makes their beliefs irrational and contradictory, instead of simply unreasonable.
Evil is logically inconsistent with the existence of an omnipotent and omni benevolent God. If we define "evil" and "good" as opposites, and determine that good will always try to destroy evil, we can conclude that an all-powerful and good being would get rid of all evil; since this is not the case, an omnipotent good being cannot exist.
An adequate solution to the problem of evil is a change in the fundamental definition of either "good", "evil", or "God". Some people choose not to believe in the omnipotence of God; others argue that evil does not really exist. This does not actually solve anything, as the rejection of a premise often turns out to be false, and the theologian who does so actually restates said premise later in the argument.
There are also many fallacious arguments that claim to retain all of the aforementioned premises and still solve the problem of evil. These are mistaken - often because the arguer implicitly rejects on of the premises, which renders the proof illogical.
One of these arguments claims that good cannot exist without evil. This rejects the premise of God's omnipotence, for an all powerful being has no limits on what it can do, including creating good without evil. It also changes the opposition of "good" and "evil" - in our definition, good tries to eliminate evil - and makes them out to be relative qualities. If something can be called good or evil without a comparison to something else, then good can exist without evil. This would only work if we defined evil as a lack of good. Moreover, even if good couldn't exist without evil, that gives no explanation as to the overwhelming amount of evil present, as good could then logically exist with just a bit of evil thrown in for comparison's sake.
