Use the properties of logarithms to rewrite each expression as the logarithm of

Tahmid Knox 2021-09-23 Answered
Use the properties of logarithms to rewrite each expression as the logarithm of a single expression. Be sure to use positive exponents and avoid radicals.

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Expert Answer

Aniqa O'Neill
Answered 2021-09-24 Author has 23240 answers
Using the above properties of logarithm, we get
\(\displaystyle{2}{\ln{{4}}}{x}^{{3}}+{3}{\ln{{y}}}-{\frac{{{1}}}{{{3}}}}{{\ln{{z}}}^{{6}}}\)
\(\displaystyle={{\ln{{\left({4}{x}^{{3}}\right)}}}^{{2}}+}{{\ln{{\left({y}\right)}}}^{{3}}-}{{\ln{{\left({z}^{{6}}\right)}}}^{{\frac{{{1}}}{{{3}}}}}}\)
\(\displaystyle={\ln{{16}}}{x}^{{6}}+{{\ln{{y}}}^{{3}}-}{{\ln{{z}}}^{{2}}}\)
\(\displaystyle={\ln{{16}}}{x}^{{6}}+{\ln{{\frac{{{y}^{{3}}}}{{{z}^{{2}}}}}}}\)
\(\displaystyle={\ln{{\left({16}{x}^{{6}}\times{\frac{{{y}^{{3}}}}{{{z}^{{2}}}}}\right)}}}\)
\(\displaystyle={\ln{{\frac{{{16}{x}^{{6}}{y}^{{3}}}}{{{z}^{{2}}}}}}}\)
\(\displaystyle\therefore{2}{\ln{{4}}}{x}^{{3}}+{3}{\ln{{y}}}-{\frac{{{1}}}{{{3}}}}{{\ln{{z}}}^{{6}}=}{\ln{{\frac{{{16}{x}^{{6}}{y}^{{3}}}}{{{z}^{{2}}}}}}}\)
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