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: \(\displaystyle{{\log}_{{\frac{{1}}{{4}}}}{x}}={\frac{{{\ln{{x}}}}}{{{\ln{{\left(\frac{{1}}{{4}}\right)}}}}}}\)

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: \(\displaystyle{{\log}_{{\frac{{1}}{{4}}}}{x}}={\frac{{{\ln{{x}}}}}{{{\ln{{\left(\frac{{1}}{{4}}\right)}}}}}}\)