Step 1

properties of logarithms

\(\displaystyle{{\log}_{{a}}{\left({\frac{{{x}}}{{{y}}}}\right)}}={\frac{{{{\log}_{{a}}{\left({x}\right)}}}}{{{{\log}_{{a}}{\left({y}\right)}}}}}\)

Now

\(\displaystyle{{\log}_{{a}}{\left({x}\right)}}={1.5}\)

\(\displaystyle{{\log}_{{a}}{\left({y}\right)}}={4.7}\)

Step 2

from property

\(\displaystyle{{\log}_{{a}}{\left({\frac{{{x}}}{{{y}}}}\right)}}\)

\(\displaystyle={\frac{{{{\log}_{{a}}{\left({x}\right)}}}}{{{{\log}_{{a}}{\left({y}\right)}}}}}\)

\(\displaystyle={\frac{{{1.5}}}{{{4.7}}}}\)

=0.3191

\(\displaystyle{{\log}_{{a}}{\left({\frac{{{x}}}{{{y}}}}\right)}}={0.3191}\) Answer

properties of logarithms

\(\displaystyle{{\log}_{{a}}{\left({\frac{{{x}}}{{{y}}}}\right)}}={\frac{{{{\log}_{{a}}{\left({x}\right)}}}}{{{{\log}_{{a}}{\left({y}\right)}}}}}\)

Now

\(\displaystyle{{\log}_{{a}}{\left({x}\right)}}={1.5}\)

\(\displaystyle{{\log}_{{a}}{\left({y}\right)}}={4.7}\)

Step 2

from property

\(\displaystyle{{\log}_{{a}}{\left({\frac{{{x}}}{{{y}}}}\right)}}\)

\(\displaystyle={\frac{{{{\log}_{{a}}{\left({x}\right)}}}}{{{{\log}_{{a}}{\left({y}\right)}}}}}\)

\(\displaystyle={\frac{{{1.5}}}{{{4.7}}}}\)

=0.3191

\(\displaystyle{{\log}_{{a}}{\left({\frac{{{x}}}{{{y}}}}\right)}}={0.3191}\) Answer