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 $\log(1000) = \log(10*100)=\log(10) + \log(100) = \log(10) + \log(10*10) = 
\log(10) + \log(10) + \log(10) = 3\log(10)$

Although an example can often demonstrate the point, it does not count as a proper proof. You can demonstrate why this idea holds using the fact that $\log_b x = y \iff b^y = x$.
$$\log_a b^x = y \implies a^y = b^x \implies a^{\frac{y}{x} } = b \implies \log_a b = \frac{y}{x} \implies y = x\log_a b$$
$$\log_a b^x = x\log_a b; b \in \mathbb{R^+}$$
