Please, write the logarithm as a ratio of common logarithms

Harlen Pritchard 2021-08-18 Answered
Please, write the logarithm as a ratio of common logarithms and natural logarithms.
\(\displaystyle{{\log}_{{3}}{\left({x}\right)}}\)
a) common logarithms
b) natural logarithms

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Expert Answer

Khribechy
Answered 2021-08-19 Author has 4505 answers
There is \(\displaystyle{{\log}_{{3}}{\left({x}\right)}}={{\log}_{{10}}{\left({x}\right)}}\cdot{{\log}_{{3}}{\left({10}\right)}}=\)
\(\displaystyle={\frac{{{{\log}_{{10}}{\left({x}\right)}}}}{{{{\log}_{{10}}{\left({3}\right)}}}}}=\)
\(\displaystyle={\frac{{{\log{{\left({x}\right)}}}}}{{{\log{{\left({3}\right)}}}}}}\)
Similarly, \(\displaystyle{{\log}_{{3}}{\left({x}\right)}}={{\log}_{{e}}{\left({x}\right)}}\cdot{{\log}_{{3}}{\left({e}\right)}}=\)
\(\displaystyle={\frac{{{{\log}_{{e}}{\left({x}\right)}}}}{{{{\log}_{{e}}{\left({3}\right)}}}}}=\)
\(\displaystyle={\frac{{{\ln{{\left({x}\right)}}}}}{{{\ln{{\left({3}\right)}}}}}}\)
a) common logarithm: \(\displaystyle{\frac{{{{\log}_{{10}}{\left({x}\right)}}}}{{{{\log}_{{10}}{\left({3}\right)}}}}}\)
b) natural logarithm: \(\displaystyle{\frac{{{{\log}_{{e}}{\left({x}\right)}}}}{{{{\log}_{{e}}{\left({3}\right)}}}}}\)
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