When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

Question

When are $\frac{1}{|x|^s}$ for $s>0$ and $\log|x|$ integrable near the origin? I'm reading Evans PDE and in the construction of the fundamental solution of Poisson's equation, he defines $$ \Phi(x) = C \log|x| $$ for $\mathbb{R}^2$ and a suitable $C$, and $$ \Phi(x) = C \frac{1}{|x|^{n-2}} $$ for $\mathbb{R}^n$ with $n\geq 3$ and $C=C(n)$ a suitable constant. The construction then goes about defining $\Phi * f$ for an $f \in C^2_c$. Clearly if $\Phi \in L^1_{\mathrm{loc}}$, then the convolution will be finite a.e., but I worry about whether this holds because of the blow up near $0$. In general in what dimensions are $$ \log |x|,~~~~~\frac{1}{|x|^s} $$ integrable in a ball around $0$?