Taking logarithms is the inverse of raising to a power. So a natural logarithm is the inverse of raising to power e. It just is, that's how it's defined.
So means the same as .
what you're saying when you say is: "What number do I raise e to the power of in order to get y?"
So means "e to the power of the number you have to raise e to the power of to get X is X."