Is Using Tikzmarknode within Tikz Matrix possible?

You do not need tikzmark here because tikzmark is needed to make a non-node a node, but everything you are playing with here are nodes. EDIT: Fixed the directions of the arrows (big thanks to manoooh) and borrowed column 1/.append style={nodes={text width=15mm}},column 2/.append style={nodes={text width=9mm}} from Zarkos answer. (However, I disagree with replacing \pgfmathparse{f(1)}\pgfmathprintnumber{\pgfmathresult} by something using \pgfmathtruncatemacro because this won't work once you have non-integer results.)

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix}
\begin{document}
\begin{tikzpicture}[declare function={f(\x)=1/ln(2)*ln(\x);}]
\matrix[matrix of math nodes,nodes={align=center,inner sep=3pt,
   text height=1.5ex,text depth=.25ex,draw=gray!40,ultra thin},
   draw,inner sep=0pt,ampersand replacement=\&,
   column 1/.append style={nodes={text width=15mm}},  
   column 2/.append style={nodes={text width=9mm}} ] (mat1)
    at (-5,0){
   |[fill=green!40!gray]| x~\text{or}~2^{y}  \& | 
   [fill=green!40!gray]| y \\ 
     1  \&  {\pgfmathparse{f(1)}\pgfmathprintnumber{\pgfmathresult}}\\
     2  \&  {\pgfmathparse{f(2)}\pgfmathprintnumber{\pgfmathresult}}\\
     4  \&  {\pgfmathparse{f(4)}\pgfmathprintnumber{\pgfmathresult}}\\
     8  \&  {\pgfmathparse{f(8)}\pgfmathprintnumber{\pgfmathresult}}\\
     \frac{1}{2} \& {\pgfmathparse{f(.5)}\pgfmathprintnumber{\pgfmathresult}}\\
    \frac{1}{4} \& {\pgfmathparse{f(.25)}\pgfmathprintnumber{\pgfmathresult}}\\
                                    };
      \node at ([yshift=-.8cm,xshift=2cm]mat1.south) 
            [circle,draw,font=\small,inner sep=1pt,label={[red]right:Select $y$}] (1){$1$};
      \node at ([yshift=-1.5cm,xshift=.7cm]mat1.south) 
            [circle,draw,font=\small,inner sep=1pt,label={[red]right:Compute
            $x$}] (2){$2$};        
     \draw[latex-] (mat1-7-1|-mat1.south) |- (2);
     \draw[latex-] (mat1-7-2|-mat1.south) |- (1);
\end{tikzpicture}
\end{document}

off-topic (since topic is solved by @marmot answer), but it may be handy at writing of your matrices ...

• with use of column 1/.append style={nodes={text width=15mm} (and similarly for column 2) you can remove |[text width=11mm]| from all cells
• with use of \pgfmathtruncatemacro{\L}{f(<value>)}\L instead \pgfmathparse{f(.5)}\pgfmathprintnumber{\pgfmathresult} gives slightly shorter code
• for nodes below matrix i would define common style

• use of ampersand replacement=\& is not necessary

  \documentclass{article}
  \usepackage{amsmath}
  \usepackage{tikz}
  \usetikzlibrary{matrix}
  
  \begin{document}
      \begin{tikzpicture}[
  declare function = {f(\x)=1/ln(2)*ln(\x);},
       circ/.style = {circle,draw,font=\small,inner sep=1pt, % <---
                      label={[red]right:#1}}                 % <--- 
                          ]
  \matrix[matrix of math nodes,
          nodes = {align=center,inner sep=3pt,
                   text height=2ex, text depth=1ex,         % <---
                   draw=gray!40,very thin},
          draw, inner sep=0pt,
          column 1/.append style={nodes={text width=15mm}},  % <---
          column 2/.append style={nodes={text width=9mm}}    % <---
          ] (mat1)
  {
  |[fill=green!40!gray]| x\text{, or }2^{y}                  % <---
      & |[fill=green!40!gray]| y              \\             % <---
  1   & {\pgfmathtruncatemacro{\L}{f(1)}\L}   \\             % <---
  2   & {\pgfmathtruncatemacro{\L}{f(2)}\L}   \\
  4   & {\pgfmathtruncatemacro{\L}{f(4)}\L}   \\
  8   & {\pgfmathtruncatemacro{\L}{f(8)}\L}   \\
  \frac{1}{2} & {\pgfmathtruncatemacro{\L}{f(0.5)}\L}     \\
  \frac{1}{4} & {\pgfmathtruncatemacro{\L}{f(0.25)}\L}    \\
  };
  \node at ([yshift=- 8mm] mat1.south east)   [circ=Select  $y$] (c1) {1}; % <---
  \node at ([yshift=-18mm] mat1-7-2)          [circ=Compute $x$] (c2) {2}; % <---
  \draw[latex-] (mat1-7-2) |- (c1);  % <---
  \draw[latex-] (mat1-7-1) |- (c2);  % <---
      \end{tikzpicture}
  \end{document}
  

edit:

• corrected direction of arrows

• added version with use of original way of calculation of values in the second column, i.e. with \pgfmathparse{f(1)}\pgfmathprintnumber{\pgfmathresult} (as noted marmot in his answer, it enable calculation of real values), for which is defined new command \calculation :

  \documentclass[tikz, margin=3mm]{standalone}
  \usetikzlibrary{matrix}
  \usepackage{amsmath}
      \begin{tikzpicture}[
  declare function = {f(\x)=1/ln(2)*ln(\x);},
       circ/.style = {circle,draw,font=\small,inner sep=1pt,  % <---
                      label={[red]right:#1}}                  % <---
                          ]
  \newcommand\calculation[1]{\pgfmathparse{f(#1)}%
                             \pgfmathprintnumber{\pgfmathresult}}  % <---
  \matrix[matrix of math nodes,
          nodes = {text height=1.5ex, text depth=0.5ex,       % <---
                   align=center, inner ysep=4pt,
                   draw=gray!40,very thin},
          draw, inner sep=0pt,
          column 1/.append style={nodes={text width=15mm}},   
          column 2/.append style={nodes={text width=11mm}},   
          row 1/.style = {nodes={fill=green!40!gray}}         % <---
          ] (mat1)
  {
  x\text{, or }2^{y}                                          % <---
      & y                             \\                      % <---
  1   & {\calculation{1}}             \\                      % <---
  2   & {\calculation{2}}             \\
  4   & {\calculation{4}}             \\
  8   & {\calculation{8}}             \\
  \frac{1}{2} & {\calculation{0.5}}   \\
  \frac{1}{4} & {\calculation{0.25}}  \\
  };
  \node at ([yshift=- 8mm] mat1.south east)   [circ=Select  $y$] (c1) {1}; % <---
  \node at ([yshift=-18mm] mat1-7-2)          [circ=Compute $x$] (c2) {2}; % <---
  \draw[latex-] (mat1-7-2) |- (c1);  % <---
  \draw[latex-] (mat1-7-1) |- (c2);  % <---
      \end{tikzpicture}
  \end{document}
  

the result is the same as above.