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

Please, write the logarithm as a ratio of common logarithms and natural logarithms.
$$\displaystyle{{\log}_{{3}}{\left({x}\right)}}$$
a) common logarithms
b) natural logarithms

• Questions are typically answered in as fast as 30 minutes

### Plainmath recommends

• Get a detailed answer even on the hardest topics.
• Ask an expert for a step-by-step guidance to learn to do it yourself.

Khribechy
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: $$\displaystyle{\frac{{{{\log}_{{e}}{\left({x}\right)}}}}{{{{\log}_{{e}}{\left({3}\right)}}}}}$$