\documentclass{article}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
\usepackage{xcolor}
\usepackage{newunicodechar}
  \makeatletter
    \def\doxynewunicodechar#1#2{%
    \@tempswafalse
    \edef\nuc@tempa{\detokenize{#1}}%
    \if\relax\nuc@tempa\relax
      \nuc@emptyargerr
    \else
      \edef\@tempb{\expandafter\@car\nuc@tempa\@nil}%
      \nuc@check
      \if@tempswa
        \@namedef{u8:\nuc@tempa}{#2}%
      \fi
    \fi
  }
  \makeatother
  \doxynewunicodechar{⁻}{${}^{-}$}% Superscript minus
  \doxynewunicodechar{²}{${}^{2}$}% Superscript two
  \doxynewunicodechar{³}{${}^{3}$}% Superscript three

\pagestyle{empty}
\begin{document}
$gain=\frac{m0x+c0}{m1x+c1}$
\pagebreak

$gain=\frac{c0}{m1x+c1}$
\pagebreak

$\frac{m0x+c0}{c1}$
\pagebreak

$gain = a \cdot 2^{m \cdot x}$
\pagebreak

$log_{2}{10^{\frac{1}{20}}}$
\pagebreak

$LP = \frac{1\mathrm{m}}{D}$
\pagebreak

\end{document}
