\documentclass[tikz]{standalone}

\usepackage{pgfplots}

\pgfplotsset{compat=1.14}

\begin{document}

\begin{tikzpicture}
    \begin{axis}[xmin=-5, xmax=5, width=12cm, height=5cm, xlabel={$\log\omega$}, ylabel={$20\log|\underline H(\omega)|$}]
        \addplot [domain=0.00001:0.001, mark=none, samples=1000, smooth] ({ln(x)/ln(10)}, {20*ln(sqrt(1+(x*3)^2)/sqrt(1+(x*5)^2))/ln(10)});
        \addplot [domain=0:1, mark=none, samples=1000, smooth] ({ln(x)/ln(10)}, {20*ln(sqrt(1+(x*3)^2)/sqrt(1+(x*5)^2))/ln(10)});
        \addplot [domain=1:10, mark=none, samples=1000, smooth] ({ln(x)/ln(10)}, {20*ln(sqrt(1+(x*3)^2)/sqrt(1+(x*5)^2))/ln(10)});
        \addplot [domain=10:100, mark=none, samples=1000, smooth] ({ln(x)/ln(10)}, {20*ln(sqrt(1+(x*3)^2)/sqrt(1+(x*5)^2))/ln(10)});
        \addplot [domain=100:1000, mark=none, samples=100, smooth] ({ln(x)/ln(10)}, {20*ln(sqrt(1+(x*3)^2)/sqrt(1+(x*5)^2))/ln(10)});
        \addplot [domain=1000:10000, mark=none, samples=100, smooth] ({ln(x)/ln(10)}, {20*ln(sqrt(1+(x*3)^2)/sqrt(1+(x*5)^2))/ln(10)});
        \addplot [domain=10000:100000, mark=none, samples=100] ({ln(x)/ln(10)}, {20*ln(sqrt(1+(x*3)^2)/sqrt(1+(x*5)^2))/ln(10)});
    \end{axis}
\end{tikzpicture}

\end{document}
