# Please, write the logarithm as a ratio of common logarithms

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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Aubree Mcintyre
$$\displaystyle{{\log}_{{a}}{b}}={\frac{{{{\log}_{{c}}{\left({a}\right)}}}}{{{{\log}_{{c}}{\left({b}\right)}}}}}$$
$$\displaystyle{{\log}_{{a}}{b}}={y}$$
$$\displaystyle{a}^{{{{\log}_{{a}}{b}}}}={a}^{{y}}$$
$$\displaystyle{b}={a}^{{y}}$$
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, $$\displaystyle{\frac{{{{\log}_{{10}}{\left({86}\right)}}}}{{{{\log}_{{10}}{\left({5}\right)}}}}}={{\log}_{{5}}{86}}$$
$$\displaystyle{\frac{{{\ln{{\left({86}\right)}}}}}{{{\ln{{\left({5}\right)}}}}}}={{\log}_{{5}}{86}}$$