How I can show that the function $ \ln |x| \in L_{loc}^1(\mathbb{R})$ slow growing using distribution theory?

Question

We say that a continuous function $g$ is slow growing if there exist $p\geq0$ and $C>0$ such that $\dfrac{g(x)}{C(1+|x|)^p}\in L_{loc}^1(k), \forall x\in \mathbb{R^n}$ and $k$ is a compact of $\mathbb{R^n}$ . I have tried to apply this definition to show that $ ln |x| \in L_{loc}^1$ $(g:\mathbb{R}\to \mathbb{R}),x\to g(x)=\ln|x|)$ slow growing , such that I assumed that there is a compact $K=[a,b] \subset \mathbb{R}$ to show the continuity of $\ln |x|$ in $k$ , I have treated $3$ cases relate to values of $a$ and $b$ ($ a, b >0 \quad \text{and}\quad b<0, a >0 \quad \text{and} \quad a<0<b)$) ,Firstly I started with if $a , b >0 $ then $g$ is continious in the compact $k$ and $ \ln|x|\in L_{loc}^1([a,b])$ the same things with other cases , but my problem with the second part how to prove existence of $p$ and $C$ for which $\dfrac{g(x)}{C(1+|x|)^p}\in L_{loc}^1(k)$ ?