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I was reading The TexBook and came to the following part on page 169:

Besides these cases, you will occasionally encounter formulas in which the symbols are bunched up too tightly, or where too much white space appears, because of certain unlucky combinations of shapes. [...] Here are some examples of situations to look out for:

$\sqrt{\,\log x}$

The positive thin space in $\sqrt{\,\log x}$ compensates for the fact that log x begins with a tall, unslanted letter

However on page 163 Knuth uses $\sqrt{{\rm Var}(X)}$, where in my opinion the rule should apply as well.

Which variant should be used?

What about upright math fonts like eulervm? Should I always write $\sqrt{\,k}$?

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I think it's always a judgment question. You should note that “log” has a descender, so the square root sign will be lowered to try and cover it: compare lines 1 and 2 in the following example, but also line 3. For the case “Var(x)”, the parentheses already provide some more room and \, doesn't seem necessary.

\halign{#\hfil&\qquad#\hfil\cr
  No {\tt\string\,}&With {\tt\string\,}\cr
  $\sqrt{\log x}$&$\sqrt{\,\log x}$\cr
  $\sqrt{\mathop{\rm Var}x}$&$\sqrt{\,\mathop{\rm Var}x}$\cr
  $\sqrt{\mathop{\rm Var}y}$&$\sqrt{\,\mathop{\rm Var}y}$\cr
  $\sqrt{{\rm Var}(x)}$&$\sqrt{\,{\rm Var}(x)}$\cr
}

\bye

enter image description here

Here's the example with eulervm:

\documentclass{article}
\usepackage{eulervm}

\begin{document}

$\sqrt{k}$ or $\sqrt{\,k}$

$\sqrt{l}$ or $\sqrt{\,l}$

$\sqrt{p}$ or $\sqrt{\,p}$

$\sqrt{K}$ or $\sqrt{\,K}$

$\sqrt{L}$ or $\sqrt{\,L}$

$\sqrt{P}$ or $\sqrt{\,P}$

\end{document}

Judge for yourself. In the “l” case, I'd probably also try \sqrt{l\,}

enter image description here

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