\documentclass[tikz]{standalone}

\usepackage{pgfplots}

\pgfplotsset{compat=1.14}

\begin{document}

\begin{tikzpicture}
    \begin{axis}[xmin=-3, xmax=9, width=12cm, height=5cm, xlabel={$\log\omega$}, ylabel={$|\underline H(\omega)|$}]
        \addplot [domain=0:1, mark=none, samples=1000] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=1:10, mark=none, samples=300] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=10:100, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=100:1000, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=1000:10000, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=10000:100000, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=100000:1000000, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=1000000:10000000, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=10000000:100000000, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
        \addplot [domain=100000000:1000000000, mark=none, samples=100] ({ln(x)/2.302585}, {x/150 * 1/ sqrt(((1-(x/1500)^2)^2+(x/150)^2))});
    \end{axis}
\end{tikzpicture}

\end{document}
