## Precalculus: Mathematics for Calculus, 7th Edition

$\log_{a}(\dfrac{x^{2}}{yz^{3}})=2\log_{a}x-\log_{a}y-3\log_{a}z$
$\log_{a}(\dfrac{x^{2}}{yz^{3}})$ The logarithm of a division can be expanded as a substraction. Like this: $\log_{a}(\dfrac{x^{2}}{yz^{3}})=\log_{a}(x^{2})-\log_{a}(yz^{3})=...$ The logarithm of a product can be expanded as a sum. We can use this law to expand $\log_{a}(yz^{3})$: $...=\log_{a}(x^{2})-[\log_{a}y+\log_{a}(z^{3})]=...$ $...=\log_{a}(x^{2})-\log_{a}y-\log_{a}(z^{3})=...$ We can take the exponents present in $\log_{a}(x^{2})$ and $\log_{a}(z^{3})$ to the front of their respective logarithms to multiply: $...=2\log_{a}x-\log_{a}y-3\log_{a}z$