Is this a proof of "$\log(a^x) = x\log(a)$"? I'm not sure if it's right, but could this be proof of "$\log(a^x) = x\log(a)$"?
For example $\log(1000) = \log(10*100)=\log(10) + \log(100) = \log(10) + \log(10*10) = 
\log(10) + \log(10) + \log(10) = 3\log(10)$

This is not a proof, your statement is an example. In order to prove a relationship like
\begin{equation}
\tag{1}
\label{1}
\log(a^x) = x\log(a)
\end{equation}
you have to take into account all possible values of $x$ and $a$ for which \ref{1} should hold. A simple version of the proof you are looking for can be done by using your idea. 
Let $x \in \mathbb{N}, a \in (0,\infty)$. By applying the log-identity $\log(a\cdot b) = \log(a) + \log(b)$ repeatedly we obtain:
\begin{align}
\log(a^x) = \log(a \cdot \dots \cdot a) = \log(a) + \log(a \cdot \dots \cdot a) = x\log(a)
\end{align}
An example can suffice as a proof, if it is in the form of a counterexample. Consider the following statement.
$$0 < \log(a) \quad \text{for all } a \in [1,\infty)$$
I can prove this statement false by providing the counterexample $a = 1$.
