Suppose you have an exponential equation like:
The "logarithm" function says: "What number do I have to raise to so as to get ?"
So we have .
Now when the number being raised to the power is (which is of course 2.71828 ...), the same thing applies.
So is another way of saying . And a special symbol for is .
Let's look at , where we're trying to find .
This is saying: "What number do I raise to so as to get ?"
What number do I raise to (where is any number) to get ?
As you can see, "undoes" the work that " to the power of" does.
That is, logarithm and exponential (if it's the same base) are inverse functions.
Now the second one is trickier.
The thing here is to remember that (it just is, the book probably shows why).
So (from what we did before).