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
ANSWERED
asked 2021-07-31
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. \(\displaystyle{{\log}_{{{4}}}{\left({x}+{6}\right)}}+{7}{{\log}_{{{4}}}{x}}\)

Expert Answers (1)

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
\(\displaystyle\Rightarrow{{\log}_{{{4}}}{\left({x}+{6}\right)}}-{{\log}_{{{4}}}{\left({x}^{{{7}}}\right)}}\)
\(\displaystyle\Rightarrow{{\log}_{{{4}}}{\left({x}+{6}\right)}}-{7}{{\log}_{{{4}}}{\left({x}\right)}}\)
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