# How do you expand log_3((uv^2)/w)?

May 17, 2016

So ${\log}_{3} \left(\frac{u {v}^{2}}{w}\right) \to \frac{{\log}_{3} \left(u\right) + 2 {\log}_{3} \left(v\right)}{\log} _ 3 \left(w\right)$

#### Explanation:

If source values are multiplied the logs are added
$\text{ } \log \left(a b\right) \to \log \left(a\right) + \log \left(b\right)$

Is source values are divided the logs are subtracted.
$\text{ } \log \left(\frac{a}{b}\right) \to \log \left(a\right) - \log \left(b\right)$

If source values are raised to some power then the log is multiplied by the value of the power
$\text{ } \log \left({a}^{4}\right) \to 4 \log \left(a\right)$

If source value are rooted then the log is divided by the value of that root
$\text{ } \log \left(\sqrt[5]{a}\right) \to \frac{1}{5} \log \left(a\right)$
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So ${\log}_{3} \left(\frac{u {v}^{2}}{w}\right) \to \frac{{\log}_{3} \left(u\right) + 2 {\log}_{3} \left(v\right)}{\log} _ 3 \left(w\right)$