Let $x\in\mathbb{R},x>0$. Prove that there exists one and only $n\in\mathbb{Z}$ so that $10^{n}\leq x<10^{n+1}$.
I thought about using the theorem that for every $x$: $a\leq x <a+1$ when $a\in \mathbb Z$ and then by using logarithm rules get to the inequality in the question, but couldn't get to it. Also how would I prove that there is only one $n$ for this?