The so-called Symm's integral equation on an interval $[a,b]$ is defined by $$\int_a^bu(y)\log|x-y|dy=f(x),\,\,x\in[a,b],$$ and $f$ is a given function.

In the introduction of a paper by I. H. Sloan and E. P. Stephan on Symm's integral equations it is stated that

"For $b-a\not= 4$ and $f$ suitably smooth, the solution to the Symm's equation is unique with end-point singularities of the form $\frac{1}{\sqrt{(x-a)(b-x)}}.$"

They reference to $\textit{ Lineare IntegralOperatoren, (Teubner-Verlag, Stuttgart, 1970)}$ by K. Jorgens, which I do not have access to. Here are my questions:

$\textbf{Q1.}$ I am very suspicious to their claim. One can take $u$ to be polynomial and after (tedious) computations you can see that $\int_a^bu(x)\log|x-y|dy=Q(x,\log x)$, where $Q$ is a polynomial of two variable and clearly $\lim_{x\to0^+}Q(x,\log x)=0$. No singularities what so ever. Is that a misunderstanding or what is the catch in here?

$\textbf{Q2.}$ Assuming $f$ is sufficiently smooth, what $b-a$ being equal to 4 has to do with end-point singularity or is that "magical" 4 tied to uniqueness?

$\textbf {Remark:}$ The computations provided by Christian Remling answers this question completely. I need to add that T. Carleman in his paper (1922) gives the solution to this integral equation as follows:

$u(x)=-\frac{1}{\sqrt{(x-a)(b-x)}}\int_a^b\frac{\sqrt{(y-a)(b-y)}f'(y)dy}{y-x}-\frac{1}{\sqrt{(x-a)(b-x)}}\frac{1}{\log(b-a)-2\log2}\int_a^b\frac{f(y)dy}{\sqrt{(y-a)(b-y)}}$

as long as $b-a\not=4$.