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
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.)

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:
$$\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}={{\log}_{{{d}}}{\left({u}^{{{8}}}\right)}}+{{\log}_{{{d}}}{\left({v}^{{{3}}}\right)}}$$
$$\displaystyle{{\log}_{{{d}}}{\left({u}^{{{8}}}{v}^{{{3}}}\right)}}={8}{{\log}_{{{d}}}{\left({u}\right)}}+{3}{{\log}_{{{d}}}{\left({v}\right)}}$$