Question

Expand using logarithmic properties. Where possible, evaluate logarithmic expressions.

Transformation properties
ANSWERED
asked 2021-08-17
Expand using logarithmic properties. Where possible, evaluate logarithmic expressions.
\(\displaystyle{{\log}_{{{5}}}{\left({\frac{{{x}^{{{3}}}\sqrt{{{y}}}}}{{{125}}}}\right)}}\)

Expert Answers (1)

2021-08-18
Step 1
On simplification, we get
\(\displaystyle{{\log}_{{{5}}}{\left({\frac{{{x}^{{{3}}}\sqrt{{{y}}}}}{{{125}}}}\right)}}={{\log}_{{{5}}}{\left({125}\right)}}\ {\left[\because{{\log}_{{{a}}}{\left({\frac{{{m}}}{{{n}}}}\right)}}={{\log}_{{{a}}}{\left({m}\right)}}-{{\log}_{{{a}}}{\left({n}\right)}}\right]}\)
\(\displaystyle={{\log}_{{{5}}}{\left({x}^{{{3}}}\right)}}+{{\log}_{{{5}}}{\left(\sqrt{{{y}}}\right)}}-{{\log}_{{{5}}}{\left({5}^{{{3}}}\right)}}\ {\left[\because{{\log}_{{{a}}}{\left({m}{n}\right)}}={{\log}_{{{a}}}{\left({m}\right)}}+{{\log}_{{{a}}}{\left({n}\right)}}\right]}\)
\(\displaystyle={{\log}_{{{5}}}{\left({x}^{{{3}}}\right)}}+{{\log}_{{{5}}}{\left({y}^{{\frac{{1}}{{2}}}}\right)}}-{{\log}_{{{5}}}{\left({5}^{{{3}}}\right)}}\)
\(\displaystyle={3}{{\log}_{{{5}}}{\left({x}\right)}}+{\frac{{{1}}}{{{2}}}}{{\log}_{{{5}}}{\left({y}\right)}}-{3}{{\log}_{{{5}}}{\left({5}\right)}}\ {\left[\because{{\log}_{{{a}}}{\left({m}^{{{n}}}\right)}}={n}{{\log}_{{{a}}}{\left({m}\right)}}\right]}\)
\(\displaystyle={3}{{\log}_{{{5}}}{\left({x}\right)}}+{\frac{{{1}}}{{{2}}}}{{\log}_{{{5}}}{\left({y}\right)}}-{3}{\left({1}\right)}\ {\left[\because{{\log}_{{{a}}}{\left({a}\right)}}={1}\right]}\)
\(\displaystyle={3}{{\log}_{{{5}}}{\left({x}\right)}}+{\frac{{{1}}}{{{2}}}}{{\log}_{{{5}}}{\left({y}\right)}}-{3}\)
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