In order for the automatic resizing of delimiters to work, nath redefines a lot of TeX internals. amsmath also defines a lot of TeX internals. Neither nath nor amsmath take into consideration that some other package might redefine the internal macros, and hence the error.
By default, \log is defined as follows:
> \log=macro:
->\mathop {\operator@font log}\nolimits .
l.3 \show\log
But, after loading amsopn (which is loaded by amsmath), we get
> \log=macro:
->\qopname \relax o{log}.
l.5 \show\log
This is the reason that the following minimal document fails:
\documentclass{article}
\usepackage{nath}
\usepackage{amsopn}
\begin{document}
$\log$
\end{document}
So, you must choose between loading amsmath and nath. For example, your minimal example can be processes without loading amsmath:
\documentclass{article}
\usepackage{nath}
\makeatletter
\def\argmin{\mathop{\operator@font arg\,min}\nolimits}
\makeatother
\begin{document}
\begin{equation}
\argmin \log( \sum_{i=1}^2 x^2)
\end{equation}
\end{document}
Another option is to save and restore the meaning of operators like \log, \max,\arg, \sin, etc.
\documentclass{article}
\let\normallog \log
\usepackage{amsmath,nath}
\let\log \normallog
\makeatletter
\def\argmin{\mathop{\operator@font arg\,min}\nolimits}
\makeatother
\begin{document}
\begin{equation}
\argmin \log( \sum_{i=1}^2 x^2)
\end{equation}
\end{document}
nath provides some macros for display math (\wall .. \return); some of the other packages that are part of amsmath bundle (amsgen, amstext, amsfonts) work with nath.
EDIT: It seems that nath is at fault here. nath redefines \mathop. Quoting from nath.sty:
\mathop is redefined to stop misinterpretation of following Bins as unary operators (cf. TeXbook, p. 170).
The following definition determines spacing between Op and Bin.
(According to [TeXBook, p. 170], ``such case never arises,
so plain TeX leaves it undefined, making the Bin into unary.'')
If you simply restore the definition of \mathop, then everything works fine:
\documentclass{article}
\usepackage{amsmath,nath}
\makeatletter
\let\mathop\o@mathop
\makeatother
\DeclareMathOperator{\argmin}{arg\,min}
\begin{document}
\begin{equation}
\argmin \log(\sum_{x=1}^2 x^2)
\end{equation}
\end{document}
nathpackage. Or rather, why are you using it? – Werner Dec 06 '13 at 04:32nath? – user541686 Dec 06 '13 at 04:41\left...\right. Auto-sizing is not always the best... use the\big-family rather (also supplied byamsmath). – Werner Dec 06 '13 at 04:56\leftand\right, that was left over from my testing... will remove. – user541686 Dec 06 '13 at 04:57