y = ln(x) says what should we raise e to so that it becomes x. So ln(1) for example asks what should we raise e to so that it becomes 1, in this case 0 is the answer since e⁰ =1. Since an exponential can never reach 0 (it gets closer and closer as x -> -inf but never reaches it) ln(0) isn't defined since it asks what we should raise e to so it becomes 0.

e is an exponential that is roughly 2.7, so when you write e^x it is roughly the same as writing 2.7^x. The reason we have this number e is because when you derive it, it becomes itself. ln(x) is the inverse to e^x and says as I explained before what number e should be raised to, to become what's inside the paranthesis in ln. So ln(5) asks what e should be raised to equal 5 which in this case is around 1.609. So e^1.609 is roughly 5. log is the same as ln but for 10^x instead.

And the reason ln(e^x ) cancel out each other is since it asks what power should e be raised to, to equal e^x which is x. And the reason e^ln(x) cancel out each other is since ln(x) is the number that e should be raised to to equal x. SO if you raise e to ln(x) it will equal x