# If \log_a(x)=1.5 and \log_a(y)=4.7 find the following u

If
$$\displaystyle{{\log}_{{a}}{\left({x}\right)}}={1.5}$$
and $$\displaystyle{{\log}_{{a}}{\left({y}\right)}}={4.7}$$
find the following using the properties of logarithms.
$$\displaystyle{{\log}_{{a}}{\left({\frac{{{x}}}{{{y}}}}\right)}}$$

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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