Question

Write the following expression as a sum and/or difference of logarithms. Express powers as factors. \log_{d}(u^{8}v^{3}) u>0, v>0 \log_{d}(u^{8}v^{3}) = ?

Factors and multiples
ANSWERED
asked 2021-08-05
Write the following expression as a sum and/or difference of logarithms. Express powers as factors.
\(\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}{u}{>}{0},{v}{>}{0}\)
\(\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}=?\) (Simplify your answer.)

Expert Answers (1)

2021-08-06
Step 1
It is required to write the expression as a sum or difference of logarithms.
The given expression is: \(\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}\)
Step 2
Use property: \(\displaystyle{{\log}_{{{d}}}{\left({a}\cdot{b}\right)}}={{\log}_{{{d}}}{\left({a}\right)}}+{{\log}_{{{d}}}{\left({b}\right)}}\)
Step 3
Now, apply the above property to simplify:
\(\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}={{\log}_{{{d}}}{\left({u}^{{{8}}}\right)}}+{{\log}_{{{d}}}{\left({v}^{{{3}}}\right)}}\)
Step 4
Now use the property: \(\displaystyle{{\log}_{{{d}}}{\left({x}^{{{n}}}\right)}}={n}{{\log}_{{{d}}}{\left({x}\right)}}\)
Step 5
By applying the property expression reduces to:
\(\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}={{\log}_{{{d}}}{\left({u}^{{{8}}}\right)}}+{{\log}_{{{d}}}{\left({v}^{{{3}}}\right)}}\)
\(\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}={8}{{\log}_{{{d}}}{\left({u}\right)}}+{3}{{\log}_{{{d}}}{\left({v}\right)}}\)
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