Question

Write the expression as a sum and/or difference of logarithms. Express powers as factors \log[\frac{x(x+4)}{(x+2)^{5}}], x>0 \log[\frac{x(x+4)}{(x+2)^{5}}]=?

Factors and multiples
Write the expression as a sum and/or difference of logarithms. Express powers as factors.
$$\displaystyle{\log{{\left[{\frac{{{x}{\left({x}+{4}\right)}}}{{{\left({x}+{2}\right)}^{{{5}}}}}}\right]}}},{x}{>}{0}$$
$$\displaystyle{\log{{\left[{\frac{{{x}{\left({x}+{4}\right)}}}{{{\left({x}+{2}\right)}^{{{5}}}}}}\right]}}}=?$$ (Simplify your answer.)

2021-08-03
Step 1
It is required to simplify the given expression:
$$\displaystyle{\log{{\left[{\frac{{{x}{\left({x}+{4}\right)}}}{{{\left({x}+{2}\right)}^{{{5}}}}}}\right]}}}$$
Step 2
Apply these properties of logarithm to simplify:
$$\displaystyle{\log{{\left({\frac{{{a}}}{{{b}}}}\right)}}}={\log{{\left({a}\right)}}}-{\log{{\left({b}\right)}}}$$
$$\displaystyle{\log{{\left({a}\cdot{b}\right)}}}={\log{{\left({a}\right)}}}+{\log{{\left({b}\right)}}}$$
$$\displaystyle{\log{{\left({x}^{{{n}}}\right)}}}={n}{\log{{\left({x}\right)}}}$$
Step 3
Now, by using these properties simplify the given expression:
$$\displaystyle{\log{{\left({\frac{{{x}{\left({x}+{4}\right)}}}{{{\left({x}+{2}\right)}^{{{5}}}}}}\right)}}}={\log{{\left({x}{\left({x}+{4}\right)}\right)}}}-{\log{{\left({\left({x}+{2}\right)}^{{{5}}}\right)}}}$$
$$\displaystyle={\log{{\left({x}\right)}}}+{\log{{\left({x}+{4}\right)}}}-{5}{\log{{\left({x}+{2}\right)}}}$$