Using the above properties of logarithm, we get

\(\displaystyle{2}{\ln{{4}}}{x}^{{3}}+{3}{\ln{{y}}}-{\frac{{{1}}}{{{3}}}}{{\ln{{z}}}^{{6}}}\)

\(\displaystyle={{\ln{{\left({4}{x}^{{3}}\right)}}}^{{2}}+}{{\ln{{\left({y}\right)}}}^{{3}}-}{{\ln{{\left({z}^{{6}}\right)}}}^{{\frac{{{1}}}{{{3}}}}}}\)

\(\displaystyle={\ln{{16}}}{x}^{{6}}+{{\ln{{y}}}^{{3}}-}{{\ln{{z}}}^{{2}}}\)

\(\displaystyle={\ln{{16}}}{x}^{{6}}+{\ln{{\frac{{{y}^{{3}}}}{{{z}^{{2}}}}}}}\)

\(\displaystyle={\ln{{\left({16}{x}^{{6}}\times{\frac{{{y}^{{3}}}}{{{z}^{{2}}}}}\right)}}}\)

\(\displaystyle={\ln{{\frac{{{16}{x}^{{6}}{y}^{{3}}}}{{{z}^{{2}}}}}}}\)

\(\displaystyle\therefore{2}{\ln{{4}}}{x}^{{3}}+{3}{\ln{{y}}}-{\frac{{{1}}}{{{3}}}}{{\ln{{z}}}^{{6}}=}{\ln{{\frac{{{16}{x}^{{6}}{y}^{{3}}}}{{{z}^{{2}}}}}}}\)

\(\displaystyle{2}{\ln{{4}}}{x}^{{3}}+{3}{\ln{{y}}}-{\frac{{{1}}}{{{3}}}}{{\ln{{z}}}^{{6}}}\)

\(\displaystyle={{\ln{{\left({4}{x}^{{3}}\right)}}}^{{2}}+}{{\ln{{\left({y}\right)}}}^{{3}}-}{{\ln{{\left({z}^{{6}}\right)}}}^{{\frac{{{1}}}{{{3}}}}}}\)

\(\displaystyle={\ln{{16}}}{x}^{{6}}+{{\ln{{y}}}^{{3}}-}{{\ln{{z}}}^{{2}}}\)

\(\displaystyle={\ln{{16}}}{x}^{{6}}+{\ln{{\frac{{{y}^{{3}}}}{{{z}^{{2}}}}}}}\)

\(\displaystyle={\ln{{\left({16}{x}^{{6}}\times{\frac{{{y}^{{3}}}}{{{z}^{{2}}}}}\right)}}}\)

\(\displaystyle={\ln{{\frac{{{16}{x}^{{6}}{y}^{{3}}}}{{{z}^{{2}}}}}}}\)

\(\displaystyle\therefore{2}{\ln{{4}}}{x}^{{3}}+{3}{\ln{{y}}}-{\frac{{{1}}}{{{3}}}}{{\ln{{z}}}^{{6}}=}{\ln{{\frac{{{16}{x}^{{6}}{y}^{{3}}}}{{{z}^{{2}}}}}}}\)