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

Please, write the logarithm as a ratio of common logarithms and natural logarithms.
$$\displaystyle\frac{{\log}_{{1}}}{{4}}{\left(х\right)}$$
a) common logarithms
b) natural logarithms

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Szeteib
a) Use the change-of-base formula using base 10
common logarithm: $$\displaystyle{{\log}_{{a}}{x}}={\frac{{{{\log}_{{10}}{x}}}}{{{{\log}_{{10}}{\left({a}\right)}}}}}$$
answer: $$\displaystyle{{\log}_{{\frac{{1}}{{4}}}}{x}}={\frac{{{{\log}_{{10}}{x}}}}{{{{\log}_{{10}}{\left(\frac{{1}}{{4}}\right)}}}}}$$
b) Use the change-of-base formula using base e
natural logarithm: $$\displaystyle{{\log}_{{a}}{x}}={\frac{{{\ln{{x}}}}}{{{\ln{{a}}}}}}$$
answer: $$\displaystyle{{\log}_{{\frac{{1}}{{4}}}}{x}}={\frac{{{\ln{{x}}}}}{{{\ln{{\left(\frac{{1}}{{4}}\right)}}}}}}$$