# How do you expand log_5 ((xy^3)/(z^4))?

May 31, 2016

${\log}_{5} \left(\frac{x {y}^{3}}{z} ^ 4\right) = {\log}_{5} \left(x\right) + 3 {\log}_{5} \left(y\right) - 4 {\log}_{5} \left(z\right)$

#### Explanation:

Source numbers multiplied $\to$ add logs
Source number to power $\to$ power x log of number
Source numbers divide $\frac{a}{b} \to \log \left(a\right) - \log \left(b\right)$

Using the above and not changing the base of the log:

${\log}_{5} \left(\frac{x {y}^{3}}{z} ^ 4\right) = {\log}_{5} \left(x\right) + 3 {\log}_{5} \left(y\right) - 4 {\log}_{5} \left(z\right)$