Please, write the logarithm as a ratio of common logarithms

opatovaL 2021-08-19 Answered
Please, write the logarithm as a ratio of common logarithms and natural logarithms.
\(\displaystyle{{\log}_{{5}}{\left({86}\right)}}\)
a) common logarithms
b) natural logarithms

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Expert Answer

Aubree Mcintyre
Answered 2021-08-20 Author has 8851 answers
\(\displaystyle{{\log}_{{a}}{b}}={\frac{{{{\log}_{{c}}{\left({a}\right)}}}}{{{{\log}_{{c}}{\left({b}\right)}}}}}\)
\(\displaystyle{{\log}_{{a}}{b}}={y}\)
\(\displaystyle{a}^{{{{\log}_{{a}}{b}}}}={a}^{{y}}\)
\(\displaystyle{b}={a}^{{y}}\)
Take \(\displaystyle{{\log}_{{c}}}\) of both sides
\(\displaystyle{{\log}_{{c}}{b}}={{\log}_{{c}}{a}^{{y}}}\)
\(\displaystyle{{\log}_{{c}}{b}}={y}{{\log}_{{c}}{a}}\)
\(\displaystyle{\frac{{{{\log}_{{c}}{\left({b}\right)}}}}{{{{\log}_{{c}}{\left({a}\right)}}}}}={y}={{\log}_{{a}}{b}}\)
So, \(\displaystyle{\frac{{{{\log}_{{10}}{\left({86}\right)}}}}{{{{\log}_{{10}}{\left({5}\right)}}}}}={{\log}_{{5}}{86}}\)
\(\displaystyle{\frac{{{\ln{{\left({86}\right)}}}}}{{{\ln{{\left({5}\right)}}}}}}={{\log}_{{5}}{86}}\)
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