Since you're using Lua(La)TeX, you may want to consider employing the nifty showhyphens package. As its name suggests, this package shows all hyphenation points that TeX considers to be valid, whether or not hyphenation actually occurs or not. While this package doesn't perform precisely the task you're trying to accomplish, I believe that its output is actually even more useful than what you're after: Not only does it tell you where hyphenations occur but it also alerts you to (i) potentially invalid hyphenation points and (ii) hyphenation points that TeX might miss entirely.
Taking your three-paragraph question as an example, a simple MWE using this package might look like this:
% !TEX program = lualatex
\documentclass[letterpaper]{article}
\usepackage{fontspec,showhyphens}
\addfontfeature{ligatures=TeX}
\begin{document}
I wrote a question (Accessing pointers in LuaTeX) hoping it would
lead me to a way of reviewing the hyphenations with LuaTeX, and
without external programs. I was encouraged to write this question.
The answers to the question ``Check all hyphenations within a document''
point to external tools.
I'm hoping for a solution which writes the hyphenation to the log file
everytime a word is split across two lines, for easy reviewing.
\end{document}
and would produce the following output:

Even though this short text contains only one hyphenated word, the little red markers show one incorrect hyphenation point (Lu-a-TeX) and one missed hyphenation point (in the word hyphenation
, and in that word's plural form). To fix these issues, it suffices to insert the command
\hyphenation{Lua-TeX hy-phen-a-tion hy-phen-a-tions}
in the preamble of the MWE, leading to the following, more satisfactory output:

Of course, TeX's hyphenation algorithm isn't perfect (nor does anyone claim that it is!), so for "real" production work you should probably want to load a custom hyphenation exception list to treat words that you (or others) have noticed in the past are not hyphenated properly. One such list, aimed at US English documents, is in the file ushyphex.tex that's available on the CTAN; it contains about 1300 words. [Interestingly, the list in this file does not (yet) include treatments of the words LuaTeX
and hyphenation(s)
!] Of course, the recommendation to load hyphenation exception lists applies to other languages as well, not just to (US) English.