The first one's giving me a headache, so I'll waffle at the second.

By definition, the "taking the logarithm of" function and the "raising e to the power of" are direct inverses.

That is, by definition, , if (it's undefined otherwise).

So we have (doing the same thing to both sides of the first of the above).

Then we note that from the second of the above, and so it follows directly that:

That holds for all and so definitely holds for .