# How do you use the properties of logarithms to expand log_5 (x^2/(y^2z^3))?

Feb 24, 2017

$2 {\log}_{5} \left(x\right) - 2 \cdot {\log}_{5} \left(y\right) - 3 \cdot {\log}_{5} \left(z\right)$

#### Explanation:

Property of log 1:

${\log}_{x} \left(\frac{a}{b}\right) = {\log}_{x} \left(a\right) - {\log}_{y} \left(b\right)$

So: ${\log}_{5} \left({x}^{2} / \left({y}^{2} \cdot {z}^{3}\right)\right) = {\log}_{5} \left({x}^{2}\right) - {\log}_{5} \left({y}^{2} \cdot {z}^{3}\right)$

Property of log 2:

${\log}_{x} \left(a \cdot b\right) = {\log}_{x} \left(a\right) + {\log}_{y} \left(b\right)$

So: ${\log}_{5} \left({y}^{2} \cdot {z}^{3}\right) = {\log}_{5} \left({y}^{2}\right) + {\log}_{5} \left({z}^{3}\right)$

Property of log 3:

${\log}_{x} \left({a}^{n}\right) = n \cdot {\log}_{x} \left(a\right)$

So: ${\log}_{5} \left({x}^{2}\right) = 2 {\log}_{5} \left(x\right)$
${\log}_{5} \left({y}^{2}\right) = 2 \cdot {\log}_{5} \left(y\right)$
${\log}_{5} \left({z}^{3}\right) = 3 \cdot {\log}_{5} \left(z\right)$

log_5(x^2/(y^2*z^3))= 2log_5(x) - [ 2*log_5(y) + 3*log_5(z)] = 2log_5(x) - 2*log_5(y) - 3*log_5(z)#