I've only looked at the first one so far, but my impression is that you are almost there, except you're supposed to end with and not !

By the way I was unfamiliar with the because symbol before now, neat.

Here's how I would change your steps:

Prove that .

Proof:

Let and with .

Then

so .

Replacing n and z on the LHS according to the original equations, we obtain .

(Edited for clarity)

The second proof would be very similar to the first, but the third is a bit different.

Prove that .

Proof:

Let with .

Then by definition, .

Substituting the value for z into the original equation, we have .

We can raise each side to the power . (We are relying on the fact that for any function.)

So .

Taking the log base b of both sides, .