Question

Write as the sum and\or difference of logarithms and express powers as factors \log_{4}(\frac{x+6}{x^{7}})

Factors and multiples
Write as the sum and\or difference of logarithms. Express powers as factors.
$$\displaystyle{{\log}_{{{4}}}{\left({\frac{{{x}+{6}}}{{{x}^{{{7}}}}}}\right)}}$$
A. $$\displaystyle{{\log}_{{{4}}}{\left({x}+{6}\right)}}-{{\log}_{{{4}}}{x}}$$
B. $$\displaystyle{7}{{\log}_{{{4}}}{x}}-{{\log}_{{{4}}}{\left({x}+{6}\right)}}$$
C. $$\displaystyle{{\log}_{{{4}}}{\left({x}+{6}\right)}}-{7}{{\log}_{{{4}}}{x}}$$
D. $$\displaystyle{{\log}_{{{4}}}{\left({x}+{6}\right)}}+{7}{{\log}_{{{4}}}{x}}$$

2021-08-01
Step 1
Given, $$\displaystyle{{\log}_{{{4}}}{\left({\frac{{{x}+{6}}}{{{x}^{{{7}}}}}}\right)}}$$
Step 2
Property used:
$$\displaystyle{{\log}_{{{a}}}{\left({\frac{{{m}}}{{{n}}}}\right)}}={{\log}_{{{a}}}{m}}-{{\log}_{{{a}}}{n}}$$
$$\displaystyle{{\log}_{{{a}}}{m}^{{{n}}}}={n}{{\log}_{{{a}}}{m}}$$
Step 3
Apply the above property, we get
$$\displaystyle\Rightarrow{{\log}_{{{4}}}{\left({x}+{6}\right)}}-{{\log}_{{{4}}}{\left({x}^{{{7}}}\right)}}$$
$$\displaystyle\Rightarrow{{\log}_{{{4}}}{\left({x}+{6}\right)}}-{7}{{\log}_{{{4}}}{\left({x}\right)}}$$